The fluffiest of pop songs can attach itself to a specific event in your mind, suddenly flooding you with memories whenever you hear the tune. In this respect, relationships and breakups are a lot like records, with each memory forming a groove into your mind. Even if you're mostly over a separation, seeing your ex move on with someone new can be confusing — especially if you haven't been as lucky in love since the two of you split. Peering into a future that you no longer have with them, a future he or she is now sharing with someone else, can be painful. A part of you knows you weren't right for each other anyway, but there's another part that's stuck on a numb feeling, like a broken record. But without the rest of the melody to frame it, that feeling has no meaning. Remember that it's just a dull reminder of your past.
To extinguish the few embers that still burn inside of you, try to remember why things ended. The two of you didn't work out for a reason. Just because someone else has found happiness with your ex doesn't mean that you would have. Don't let these sudden pangs of wanting make you forget whatever happiness you've had since your breakup.
Concern yourself with you, and don't compare your life to theirs. This isn't a contest where the person who moves on first is the winner and the other is the loser. Distract yourself from these pesky feelings by taking the time to explore your interests. Dedicate yourself to an old hobby or pick up a new one — anything to get your mind off of your ex's haves and your have-nots.
Music soothes the savage beast, but hit next when a breakup song comes on. These songs may make you feel better for a little while, but they'll also keep you focused on the object of your disaffection. There's a sad irony to singing along with a tune about how over the relationship you are. Instead, scan your library for songs that make you feel generally better and direct your energy towards the present … without your ex.
Cut off contact … if you have to. If your ex's new life has deeply affected you, it's OK to remove yourself from the situation. While remaining friends with an ex is admirable, this attitude can actually be really damaging for some. You shouldn't willingly engage in a connection that causes you stress. Without the constant reminder staring you in the face, you'll be able to concentrate more on your own happiness.
What are your tips for dealing with an ex moving on? Share with fellow readers in a comment below!


text:
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Ila Garg
Kargil War Heroes – Part 12
Kargil war was a sad chapter in India's book. It resulted in a massive chaos, bloodshed, and loss of precious lives. Our brave soldiers risked their lives and put the nation before self – something they are taught during their training days.
Among the martyrs, Capt. Jerry Prem Raj was awarded Vir Chakra, the third highest military honour for his valour. He was a brave son of India.
Born in Venganoor, Trivandrum, Kerala, Capt. Jerry Prem Raj was commissioned into the 158 Medium Regiment (SP) on 5 September 1997. He was a sincere and upright officer with an eye for detail and a pleasant disposition. His father, Mr. Retna Raj worked in the Department of Technical Education, Kerala and his mother, Mrs. Chella Thayee worked in the Health Department of the government of Kerala. His brother, Rejinald Pavithran, is a retired officer of the Indian Air force.
Capt. Jerry Premraj had been married barely a month and was on his hone
ymoon when the duty called him and he had to cut short his vacation and re-join the unit in Mushkoh Valley in July 1999 for Operation Vijay. His task was to attack Twin Bumps located in Point 4875 area, Gun Hill, Drass sector. During the assault, in the wee hours of 7th July 1999, Captain Jerry Prem Raj was the Forward Observation Post Officer with 2 Naga ( A unit of Indian Army).
He searched all the enemy positions and then directed accurate artillery fire onto them which had a devastating effect on the enemy. Subsequently, he was shot and badly wounded in the enemy sniper fire. Despite injuries, he remained undeterred and continued to attack the enemy positions. Amidst the heavy fire, he displayed unparalleled grit and courage and inflicted heavy casualties on the enemy.
By then, Captain Jerry Prem Raj was severely injured. His efforts in the battle proved his mettle. He was a true war hero. He died later on, the same day. His body was cremated in his native Thiruvananthapuram. His wife gradually moved on from the truncated marriage. His memory became an occasional, but uncontrollable sob in his modest home in Venganoor.
Obviously people might forget, but the pain of the families of these martyrs remains the same. Let us not forget these valiant soldiers in our prayers.
More in this segment:
Kargil War Heroes – Part 1
Capt. Jerry Prem Raj
kargil 16 years
Kargil Heroes
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Barely visible from the sky, Norfolk Island is a world of unspoilt beauty, where a forest of evergreen pine meets the ocean, lush hills and valleys meet the beach. This is how life should be.
Part of Australia for more than 100 years, Norfolk Island was first settled in the 14th and 15th Centuries by seafarers from New Zealand – well before Captain James Cook arrived in 1774 and named the island. When the First Fleet arrived, 15 convicts and 7 free men were sent to the island to take control, with more arriving in the first 12 months. The commercial development of the island began and in the years that followed, the island was transformed into a prosperous region, predominantly through the export of the Norfolk Island Pine. Though there was development, the splendour and the beauty of the island remained intact and has been preserved since.
Just 2 hours flight from Sydney and Auckland, Norfolk Island is a tropical paradise.
The mild tropical weather conditions will have you comfortable at any time of year. The summer days are generally under 28 degrees, the nights a warm 20, while winter temperatures can be anywhere from 12 at night to 20 during the day. The island has a range of accommodation options with more than 60 properties to suit all kinds of budgets. From hotels and luxury cottages, to self contained units and budget apartments. When arriving you need to have your accommodation pre-booked or you won't even make it past immigration at the airport.
The food on Norfolk is fantastic. All fruit and vegetables are actually grown on the island, freshly picked and ready to eat!! Not to mention the fresh fish and beef, as well as locally made produce like coffee, honey and cheese. There are more than 30 restaurants, cafes and clubs where you can enjoy fantastic fresh dishes. Or if you fancy something more along the lines of some good Italian or Chinese food, you won't be disappointed either, or you might like to experience local culture with a cliff-top BBQ, sunset fish fries or a progressive dinner.
If its shopping you're after, there are markets held every Sunday morning with a range of arts and crafts, souvenirs and home made produce. Meanwhile the Farmers markets are held every Saturday morning and you can buy yourself some seasonal fruit and vegetables for a great price!
Most of the tours on the island are based around the natural environment, from 4WD eco tours to island discoveries, horse riding and reef tours. If you want to tour the island yourself, you can hire a rental car for a great price and take yourself around the island. Grab yourself a map and take a drive to some of the island's great beaches!
Norfolk Island beaches include Slaughter Bay, Anson Bay, Norfolk Island Beach, Duncombe Bay and Emily Bay. Emily Bay is the safest place on the island to swim and the sparkling crystal clear waters are protected from both the breakers and the southern ocean. Its popular for young families in particular who can rest assured their little ones will enjoy a swim in safe waters. Anson Bay is the island's most gorgeous beach and is located at the bottom of a steep cliff face. Its more secluded than the other beaches and is a great place to explore. The track down is not for the light hearted, but its well worth it once you're at the bottom. Its a popular beach for surfers and swimmers, but you need to be careful of the drop off when you're walking out into the water as it might come as a surprise. If snorkelling is what you're after, Slaughter Bay is full of colourful marine life and well worth the visit.
If you want to see the ocean from a different view, there are plenty of boat tours that will take you out to explore. To discover life under the ocean, without even getting your feet wet, take a glass bottom boat tour to see the wonderful array of tropical fish in one of the world's most southern coral reefs. You might also take to the waters in a kayak for a relaxing paddle, try your hand at fishing, or plunge right in with one of the diving or snorkelling tours.
Back on the land, you can find some fascinating tours and attractions that will take you around the island, whether you go on your own or with a tour group.
The island has a history of carnage that will have you cringing, but will also leave you fascinated, and a great walk to do is the ghost tour which will introduce you not only to stories of the past, but if you're lucky, you might even see or hear something "ghostly" yourself! Learn about the commandants, convicts and settlers who once lived on the island, which is actually known as one of the most brutal settlements of the British Empire.
Tours take you through Kingston – the main township during the Second and early Third Settlements. Kingston is in the centre of the World Heritage Listed Kingston and Arthurs Vale Historic Area – one of the group of 11 Australian Convict Sites. There are 4 museums that tell the story of the island's history and you can take a guided tour of the Area or follow the walking tracks yourself. As a place of secondary punishment, the area has a reputation as one of the harshest and cruellest penal settlement. Visit the cemetery, ruins and standing structures from centuries past, buildings from the convict era, and see the archaeological remains, landform and landscape elements. If you take the ghost tour, you will hear about the harsh history of the region from locals in the know. You might also like to take a historic tour of the old settlement sites at Pitcairn Settlers Village, visit Bloody Bridge, Captain Cook lookout and Cooks monument.
If you love getting back to nature, the national park is the place to go with its abundance of Norfolk Island's pine trees and rainforest. The National Park covers 650 hectares and comprises two sections, the Mount Pitt section on Norfolk Island 4.60 square kilometres and the neighbouring 1.9 square kilometre Phillip Island, as well as the much smaller Nepean Island. The National Park has some fantastic walking tracks, as well as stunning views from both Mt Pitt and Mt Bates – the highest points on the island. Take along a picnic or some food ready for a BBQ feast, get your walking shoes on and enjoy a journey through lush palm forests, along the coastal fringe and into a world of Norfolk Island pine. You can even walk beneath the tallest tree ferns on the planet!
There are a range of walks, from gentle to moderate and varying in length up to an hour and a half. Some of the tracks include steep sections and many of the tracks can become very slippery if its wet, so just be wary of what track you are doing on what day. The Bird Rock track has the most challenging section in the park and will take you on a steep walk, through pine forest to the cliff top overlooking Bird Rock and return. If you don't want to do the steep climb, you can take the Red Stone link track. The Old Mountain track was the main access to Mt Pitt prior to World War II and you can still see the remnants of a gun emplacement; while the 1.7 kilometre Bridle track takes you along the edge of the coastline. And easy walk is the Summit track which takes you along the ridge top from the peak of Mt Pitt to the peak of Mt Bates.
Within the park is the botanic gardens, which allow you to walk through the gorgeous flora and fauna of the region. The Botanic gardens have boardwalks suitable for wheelchairs and strollers and you can wander through a display of gorgeous plants in their natural environment, and if you're lucky you might even come across some of the island's birdlife, including the green parrot which is not found anywhere else in the world! In fact, many of the species you will find in the National Park have evolved into unique, or endemic, forms due to the isolation on Norfolk. The longest walk here is the Rainforest Gully circuit, 600 metres long, where you can wander through the rainforest gully looking in wonder at the giant vines and gorgeous surrounds.
Norfolk Island truly is a nature lover's paradise and with its unspoilt beauty, you can see why we had to include it in our Top 10 destinations.


text:
\section{Introduction} \label{sec:intro}

Hot super-Earths are a subset of the super-Earth planet population (1~R$_{\oplus }$ $<$ R$_{p}$ $<$ 2~R$_{\oplus }$), with short orbital periods (P $<$ 10 days), and surface temperatures high enough to melt silicate rock (T $>$ 800 K) due to strong irradiation by their host stars. Hot super-Earths are compelling objects to study for the insights that they provide into atmospheric loss/retention, volatile cycling, the behaviors of materials at extreme temperatures, and Earth's early history as a magma-ocean planet.

NASA\textquotesingle s \textit{Kepler} space telescope \citep{borucki2010kepler} transformed our understanding of exoplanets with discoveries of new planet classes and planetary systems. One of the \textit{Kepler} mission's most revolutionary scientific results was that among the short-period planets it was sensitive to (P $<$ 100 days), the size of the most common planet in the galaxy is between the size of Earth and Neptune (1-4~R$_{\oplus }$), which has no Solar System analog \citep{batalha2014exploring}. This population of planets is subdivided into super-Earths, 1~R$_{\oplus }$ $<$ R$_{p}$ $<$ 2~R$_{\oplus }$, and sub-Neptunes, 2~R$_{\oplus }$ $<$ R$_{p}$ $<$ 4~R$_{\oplus }$ \citep{fulton2017california}. The repurposing of the \textit{Kepler} mission into \textit{K2} \citep{howell2014k2} provided the opportunity to search for many more ultra-short period (USP) planets (P $<$ 1 day). The \textit{K2} Campaigns observed target fields in the ecliptic plane for 80 days at a time, making USP planets easily detectable within this observing window \citep{adams2016ultra, adams2017ultra, malavolta2018ultra, adams2021ultra}.

There are many unresolved theories regarding the atmospheres of hot super-Earths, including whether they exist (e.g. \citealp{kreidberg2019absence}), what they are composed of (e.g. \citealp{schaefer2009chemistry, ito2015theoretical, mansfield2019identifying}), and how they evolve. The "radius gap" -- a local minimum in the planet radius distribution at 1.75~R$_\earth$ for planets orbiting Sun-like stars, and with P $<$ 100 days \citep{fulton2017california} -- is theorized to separate predominantly rocky planets from planets with a substantial H/He atmosphere \citep{owen2017evaporation}. The location of the radius gap is dependent on the host star type, and shifts to a smaller radius as the stellar radius decreases, as seen for planets around M-dwarfs \citep{zeng2017gap, cloutier2020evolution}. \cite{rogers2015most} found that statistically planets with R$_{p} >$ 1.6~R$_\earth$ have a volatile-rich envelope.
A variety of compositions have been determined for rocky planets below the radius gap, including Earth-like compositions (e.g. \citealp{pepe2013earth, dressing2015mass}) and high density compositions akin to Mercury (e.g. \citealp{santerne2018earth, lam2021gj, silva2022hd}).

To explain the existence of the radius gap/valley, multiple theoretical models have been suggested. These include: photoevaporation -- atmospheric loss driven by stellar irradiation \citep{lopez2012thermal, chen2016evolutionary, owen2017evaporation}; core-powered mass loss -- atmospheric loss driven by the cooling of the planetary core after formation, resulting in the escape of the outer layers of the atmosphere \citep{ginzburg2018core,gupta2019sculpting,gupta2020signatures}; and gas-poor formation -- the formation of distinct rocky and non-rocky planet populations, where the rocky planet population is a result of delayed gas accretion within the protoplanetary disk until a point where the gas in the disk has almost fully dissipated \citep{lee2014make,lee2016breeding,lopez2018formation,cloutier2020evolution}.

While planets discovered by \textit{Kepler} have well-constrained radii measurements, the vast majority lack corresponding mass measurements because the planets orbit faint stars, making detailed follow-up investigations difficult. NASA\textquotesingle s Transiting Exoplanet Survey Satellite (\emph{TESS}; \citealp{ricker2014transiting}) mission, the successor of \textit{Kepler}, is an all-sky survey of bright, nearby stars, with a minimum observing baseline of $\sim27$ days. \emph{TESS}\ has identified hundreds of short-period super-Earth planet candidates amenable to follow-up observations with radial velocity (RV) instruments to determine their masses since it began operations in 2018 \citep{guerrero2021tess}. Here we present the discovery and confirmation of TOI-1075~b, an ultra-short period super-Earth around TOI-1075\ ($V = 12.75$~mag) monitored by \emph{TESS}, with a planetary radius located slightly above the radius gap. Obtaining precise radii and masses for the small, close-in, \emph{TESS}\ planet candidates that span the radius valley is crucial for elucidating the atmospheric composition and evolution of hot super-Earths via further spectroscopic characterization, and for furthering our understanding of planetary compositions by studying planetary system architectures and formation histories.\\

This paper is structured as follows. In Section \ref{sec:stellardata}, we describe the properties of the host star TOI-1075. In Section \ref{sec:planetdata}, we describe the time-series photometry and RV data sets we obtained for the TOI-1075\ system. In Section \ref{sec:juliet_jointfit}, we describe our data analysis, including a global model fit, and derive properties for the planetary system. In Section \ref{sec:discussion}, we discuss the new star-planet system, including atmospheric characterization prospects and a review of potential formation mechanisms for TOI-1075~b, and finally we provide our conclusions in Section \ref{sec:conclusions}.

\section{Stellar Data \& Characterization \label{sec:stellardata}}

\subsection{Astrometry \& Photometry}

Stellar astrometry and visible and infrared photometry for TOI-1075\ (TIC~351601843, 2MASS~J20395334-6526579, APASS~31990797, Gaia~DR3~6426188308031756288, UCAC4~123-179251) are compiled in Table \ref{tab:star}.
The positions, proper motions, parallax, radial velocity, and Gaia photometry are from Gaia DR3 \citep{prusti2016gaia, GaiaDR3}.
We convert the astrometry to Galactic velocities following
\cite{ESA1997}\footnote{$U$ towards Galactic center, $V$ in direction of Galactic spin, and $W$ towards North Galactic Pole \citep{ESA1997}.}.
Photometry is reported from APASS Data Release 10 \citep{Henden2016}\footnote{https://www.aavso.org/apass}, the \emph{TESS}\ Input Catalog (TIC8; \citealp{Stassun2019}), 2MASS \citep{Cutri2003}, and WISE \citep{Cutri2012}.
From comparison of the star's colors ($B$-$V$ = 1.42, $V$-$K_s$ = 3.58, $G$-$K_s$ = 2.93, $V$-$J$  = 2.76, $G_{Bp}$-$G_{Rp}$ = 1.84) and absolute magnitude ($M_V$ = 8.76, $M_G$ = 8.10, $M_{Ks}$ = 5.17) with typical parameters for stars of various spectral types, TOI-1075's photometry appears to be consistent with that of a typical main sequence star intermediate between K9V and M0V types \citep{Pecaut2013}.

\begin{table*}[tb]
    \caption{Astrometry \& Photometry for TOI-1075}
    \centering
    \begin{tabular}{lcc}
    \hline
    Parameter & Value & Source \\
     \hline
    Designations & TIC 351601843 & \citet{Stassun2019}\\
    RA (ICRS, J2000) & 20:39:53.082 & Gaia DR3\\
    Dec (ICRS, J2000) & -65:26:58.95 & Gaia DR3\\
    $\mu$ RA (mas yr$^{-1}$) & -99.8399 $\pm$ 0.0081 & Gaia DR3\\
    $\mu$ Dec (mas yr$^{-1}$)& -60.016 $\pm$ 0.013 & Gaia DR3\\
    Parallax (mas) & 16.2816 $\pm$ 0.0132 & Gaia DR3\\
    Distance (pc) & $61.43^{+0.18}_{-0.67}$ & Gaia DR3\\
    $v_R$ (km\,s$^{-1}$) &  31.07 $\pm$ 0.30 & Gaia DR3\\
    Spectral Type & K9V/M0V & \citet{Pecaut2013}\\
    \hline
     $B$ (mag) & 14.108 $\pm$ 0.028 & APASS/DR10\\
     $V$ (mag) & 12.751 $\pm$ 0.077 & APASS/DR10\\
     $g'$ (mag) & 13.423 $\pm$ 0.027 & APASS/DR10\\
     $r'$ (mag) & 12.181 $\pm$ 0.088 & APASS/DR10\\
     $i'$ (mag) & 11.504 $\pm$ 0.143 & APASS/DR10\\
     {\it TESS} (mag) & 10.2565 $\pm$ 0.0074 & TICv8\\
      $G$ (mag) & 12.0447 $\pm$ 0.0028 & Gaia DR3\\
      $G_{BP}$ (mag) & 12.9442 $\pm$ 0.0028 & Gaia DR3\\
      $G_{RP}$ (mag) & 11.1069 $\pm$ 0.0038 & Gaia DR3\\
      $J$ (mag) & 9.935 $\pm$ 0.023 & 2MASS\\
      $H$ (mag) & 9.292 $\pm$ 0.026 & 2MASS\\
      $K_{s}$ (mag) & 9.115 $\pm$ 0.023 & 2MASS\\
      $W_{1}$ (mag) & 9.001 $\pm$ 0.025 & WISE\\
      $W_{2}$ (mag) & 9.001 $\pm$ 0.021 & WISE\\
      $W_{3}$ (mag) & 8.915 $\pm$ 0.024 & WISE\\
      $W_{4}$ (mag) & 8.806 $\pm$ 0.315 & WISE\\
\hline
    \label{tab:star} \end{tabular}
\end{table*}

\subsection{Spectral Energy Distribution \label{sec:SED}}

As an independent determination of the basic stellar parameters, we performed an analysis of the broadband spectral energy distribution (SED) of the star together with the Gaia DR3 parallax \citep[with no systematic offset applied; see, e.g.,][]{StassunTorres:2021}, in order to determine an empirical measurement of the stellar radius and mass following the procedures described in \citet{Stassun:2016,Stassun:2017,Stassun:2018}. We pulled the $JHK_S$ magnitudes from {\it 2MASS}, the W1--W4 magnitudes from {\it WISE}, and the $G$, $G_{\rm BP}$ and $G_{\rm RP}$ magnitudes from Gaia. Together, the available photometry spans the stellar SED over the wavelength range 0.4--20~$\mu$m (see Figure~\ref{fig:sed}).

\begin{figure}[htb!]
    \centering
    \includegraphics[trim=70 80 80 80,clip,angle=90,width=\linewidth]{figures/toi_1075_sed.pdf}
\caption{Spectral energy distribution of TOI-1075. Red symbols represent the observed photometric measurements, where the horizontal bars represent the effective width of the passband. Blue symbols are the model fluxes from the best-fit NextGen atmosphere model (black).  \label{fig:sed}}
\end{figure}

We performed a fit using NExtGen stellar atmosphere models, with the free parameters being the effective temperature ($T_{\rm eff}$) and metallicity ([Fe/H]). The remaining free parameter is the extinction $A_V$, which we fixed at zero due to the star's proximity\footnote{The
STILISM 3D reddening maps from \citet{Lallement2018} estimate the reddening towards TOI-1075\ to be E(B-V) = $0.004\pm0.016$, i.e. negligible.}.
The resulting fit (Figure~\ref{fig:sed}) has a reduced $\chi^2$ of 1.4, with best fit $T_{\rm eff} = 3875 \pm 75$~K and [Fe/H] = $-0.5 \pm 0.5$. Integrating the model SED gives the observed bolometric flux, $F_{\rm bol} = 5.82 \pm 0.14 \times 10^{-10}$ erg~s$^{-1}$~cm$^{-2}$
($m_{\rm bol}$ = $11.590\pm0.026$ mag on IAU 2015 scale).
Adopting the Gaia DR3 parallax ($\varpi$ = $16.2816\pm0.0132$ mas), this leads to a bolometric luminosity of log($L_{\rm bol}$/$L_{\odot}$) = $-1.163\pm0.010$.
Combining the luminosity with the derived $T_{\rm eff}$ provides an estimate of the stellar radius of $R_\star = 0.581\pm0.024$~$R_\odot$.
In addition, we can estimate the stellar mass from the empirical $M_K$-based relations of \citet{Mann:2019}, which give $M_\star = 0.604 \pm 0.030$~M$_\odot$. Moreover, the radius and mass together imply a mean stellar density of $\rho_\star = 4.34 \pm 0.57$~g~cm$^{-3}$.

\subsection{Stellar Mass and Radius from Empirical Relations}

While we adopt the host star parameters derived above (Section \ref{sec:SED}), based on the SED and NExtGen models, we estimated those parameters using the empirical relations of \cite{Mann:2019} and \cite{Boyajian:2012}, for comparison.
We used the Gaia DR3 distance to derive the absolute $K$ band magnitude from the observed 2MASS magnitude, resulting in $M_K = 5.173 \pm 0.023$ mag.

Next, we used the empirical relation between stellar mass and $M_K$ provided by \citet[][see their Table 6 and Equation 2]{Mann:2019}. Assuming a conservative uncertainty of $5\%$ this resulted in $M_\star = 0.571 \pm 0.029\ \ensuremath{M_\sun}$. We note that the empirical relations provided by \cite{Mann:2019} cover a mass range that reaches about 0.75 \ensuremath{M_\sun}\ and a $K$ band magnitude up to about 4 mag, so the TOI 1075 host star is well within that range. For comparison, we calculated the stellar mass with the empirical relation of \citet[see their Table 1 and Equation 10]{Mann:2015}, resulting in $0.604 \pm 0.030\ \ensuremath{M_\sun}$, which is 5\% from the above estimate.

To estimate the stellar radius we used the mass derived above and the radius-mass empirical relation derived by \citet[][their Equation 10]{Boyajian:2012}, resulting in $0.541 \pm 0.027\ \ensuremath{R_\sun}$. For comparison, using the empirical relation between radius and $M_K$ of \citet[][see their Table 1]{Mann:2015}, results in $R_\star = 0.580 \pm 0.029\ \ensuremath{R_\sun}$, which is 7\% or 1.3$\sigma$ larger than the above estimate. \\

For the subsequent analyses in this paper, we adopt the stellar parameters derived from the SED analysis following \citet{Stassun:2016,Stassun:2017,Stassun:2018}, namely $R_\star = 0.581\pm0.024 R_\odot$, $M_\star = 0.604 \pm 0.030$~M$_\odot$, and $T_{\rm eff} = 3875 \pm 75$~K.

\subsection{Speckle Observations}

If a star hosting a planet candidate has a closely bound stellar companion (or companions), the companion can create a false-positive exoplanet detection if it is a stellar eclipsing binary (EB). Additionally, flux from these companion source(s) can lead to an underestimated planetary radius if not accounted for in the transit model \citep{ciardi2015understanding}. To search for close-in bound companions unresolved in our other follow-up observations, we obtained high resolution speckle imaging observations.

TOI-1075\ was observed on 2019 September 12 UT using the Zorro speckle instrument on Gemini South \citep{scott2021twin}. Zorro provides simultaneous speckle imaging in two bands (562 $\pm$ 54 nm and 832 $\pm$ 40 nm) with output data products including a reconstructed image and robust contrast limits on companion detections \citep{howell2011speckle,howell2016speckle}. Figure \ref{fig:speckle} shows the 5-sigma limiting contrast curves for the Zorro observations in both 562 nm (blue line) and 832 nm (red line), and the 832 nm reconstructed speckle image. We find that TOI-1075\ is a single star with no companion brighter than $\delta$m = 5--6 mag at 832 nm from about 0.1" out to 1.2". At the distance to TOI-1075\ ($d=61.4$~pc), these angular limits correspond to spatial separations of 6 to 74 AU.

\begin{figure}[!ht]
    \centering
    \includegraphics[width=\linewidth]{figures/TOI1075_speckle_plot.pdf}
\caption{Gemini South Zorro contrast curves for 562 nm (blue line) and 832 nm (red line) speckle observations, and 832 nm image (inset) of TOI-1075. No visual companions are detected anywhere in Zorro's field of view. \label{fig:speckle}}
\end{figure}

\subsection{Stellar Kinematics and Population}

\citet{GaiaDR3} provides the most accurate position and proper motion for TOI-1075\ (=Gaia~DR3~6426188308031756288) and \citet{BailerJones2021} converts the Gaia DR3 trigonmetric parallax ($\varpi = 16.282\pm0.013$ mas) into a geometric distance of $d = 61.43^{+0.18}_{-0.67}$ pc.
\citet{GaiaDR2} lists a median radial velocity of $v_r$ = $31.07 \pm 0.30$ km\,s$^{-1}$\, averaged over 24 epochs.
Using formulae from \citet{ESA1997}, we convert the Gaia DR3 astrometry and radial velocity into Galactic barycentric velocities:
$U, V, W$ = 33.49, -31.58, 7.49 ($\pm$0.22, 0.30, 0.20) km\,s$^{-1}$\,\footnote{The velocities are within 0.01 km\,s$^{-1}$\, of that reported in Gaia GCNS catalog \citet{GCNS2021}.}.
The star's 3D velocity is not near any of the 29 nearby young stellar groups tracked by \citet{Gagne2018}, and the BANYAN $\Sigma$ tool\footnote{http://www.exoplanetes.umontreal.ca/banyan/banyansigma.php}
returns membership probabilities of zero ($<$0.1\%).
Using the formulae and parameters from \citet{Bensby2014}, we estimate Galactic population kinematic membership probabilities for TOI-1075 of P(thin disk) = 98.2\%, P(thick disk) = 1.8\%, P(halo) = $4.2\times10^{-3}$\%, P(Hercules) = $1.3\times10^{-2}$\%, i.e. TOI-1075 is
$\sim$56$\times$ more likely to be a thin disk star than a thick disk star based on its velocity alone. %
The oldest thin disk stars are approximately $\sim$8-9 Gyr \citep[e.g.][]{Kilic2017,Fuhrmann2017,Fantin2019,Tononi2019}.

We calculated the 3D separation between TOI-1075\ and all the stars in the Gaia Catalogue of Nearby Stars \citep{GCNS2021} to search for any potential stellar companions.
We find that TOI-1075\ has no neighbors within 2\,pc,
and Gaia~DR3~6429596764016919296 ($\Delta$ = 2.00\,pc; component of the tight binary 2MASS 20335172-6403200) is its nearest star.

The systematic survey for common proper motion companions to stars within 100\,pc by \citet{Kervella2022} did not yield any matches of TOI-1075\ with any {\it Hipparcos} stars.
A query of the \citet{GaiaDR3} catalog within 2$^{\circ}$ ($\sim$2.0\,pc) of TOI-1075\ searching among stars with parallaxes and proper motions within 25\%\, of the values for TOI-1075\ yielded no plausible common proper motion companions.
Thus, TOI-1075\ appears to be a single star.\\

The space velocity of TOI-1075\ may provide some additional clues about its age.
In the GCNS catalog of stars within 100\,pc, there are 153 stars whose $UVW$ velocities are within 10 km\,s$^{-1}$\, of TOI-1075.
Thirty eight of the 153 stars are lacking SIMBAD entries and have not been noted in the literature.
Among the 115 with SIMBAD entries, 60 have fiducial spectral types in SIMBAD
and none are hotter than the F5V star HD~43879 \citep[$T_{\rm eff}$\, = $6566\pm86$ K;][]{Casagrande2011}.
A query of stellar parameter catalogs with mass estimates \citep{Schofield2019,Stassun2019,Paegert2021,Reiners2022},
shows that among the $d$ $<$ 100\,pc stars with velocities within 10\,km\,s$^{-1}$\, of TOI-1075, there is a noticeable lack of stars more massive than 1.40\,\ensuremath{M_\sun}\footnote{
HIP~29888 (HD~43879; $1.38\pm0.23$ \ensuremath{M_\sun}) in the TICv8.2 \citep{Paegert2021}, and
HIP~29888, HIP~111971 (HD~214729), and HIP~70196 (HD~125346) in \citep{Reiners2020} - all with mass estimates of
1.40 \ensuremath{M_\sun}\, - defining the upper mass envelope.}.
The list of GCNS stars with velocities within 10\,km\,s$^{-1}$\, of TOI-1075\ was also queried through the compendium of chromospheric activity measurements (\ensuremath{\log R^\prime_{\rm HK}}) from \citet{BoroSaikia2018}.
Only 8 of the stars had \ensuremath{\log R^\prime_{\rm HK}}\, measurements, and only two had \ensuremath{\log R^\prime_{\rm HK}}\, $>$ -4.8 \citep[approximately corresponding
to the Sun on its very most active days;][]{Egeland2017}: the planet host star HD~128356 (HIP~71481; K2.5IV, \ensuremath{\log R^\prime_{\rm HK}}\, = -4.73) and 6~And (HD~218804, HIP~114430; F5V, \ensuremath{\log R^\prime_{\rm HK}}\, = -4.52).
6~And is a 2.99$^{+0.48}_{-0.99}$ Gyr-old, fast-rotating \citep[$v$sin$i$ = 19\,km\,s$^{-1}$;][]{Schroeder2009} F5V star near the Kraft break, and so not unusually fast-rotating or young.
The \ensuremath{\log R^\prime_{\rm HK}}\, value for HD~128356 (-4.73) appears to be spurious, however, as the assumed B-V color from Hipparcos (0.685) is based
on a single ground-based measurement \citep{Mermilliod1997} which is at odds with the star's spectral type
\citep[K3V or K2.5IV;][]{Upgren1972,Gray2006} and the star's $T_{\rm eff}$, for which the published estimates are in tight agreement
\citep[4932-4953\,K;][]{Sousa2018,Luck2018,Soto2018}.
Adopting the B-V estimate for HD~128356 from the Tycho catalog ($1.04\pm0.02$ mag), which is more consistent with that for a K3 dwarf star, the median Mt.~Wilson S-value quoted by \citet[][S=0.214]{BoroSaikia2018} translates \citep[via formulae from ][]{Noyes1984} to a more benign chromospheric activity level of \ensuremath{\log R^\prime_{\rm HK}}\, = -5.06.
Hence, the stars that have 3D velocities within 10\,km\,s$^{-1}$\, of TOI-1075\ that are Sun-like (excluding the mid-F star 6~And) with Ca H \& K indices all have \ensuremath{\log R^\prime_{\rm HK}}\, $<$ -4.8, consistent with ages of $\gtrsim$3 Gyr \citep{Mamajek2008}.\\

We conclude that the lack of $>$1.40\,\ensuremath{M_\sun}\, stars with similar velocities to TOI-1075\ suggests that stars with similar orbits are unlikely to be $\lesssim$2 Gyr, and the small number of stars with published chromospheric activity indices seem to tell a similar story (lacking in stars $\lesssim$3 Gyr).
It appears that stars younger than $\lesssim$2 Gyr have not yet scattered into the velocity space adjacent to the orbit of TOI-1075, suggesting that the star is either a middle-aged or old thin disk star, likely with an age between 2-9 Gyr.

\subsection{Metallicity}\label{sec:metallicity}

TOI-1075\ was spectrally characterized by the RAVE \citep[RAVE~J203953.3-652658,][]{Kordopatis2013,Kunder2017,Steinmetz2020}
and GALAH \citep{Buder2018,Buder2021} stellar spectroscopy surveys, from which wildly disparate metallicity estimates have been published, ranging from [M/H] = $-0.99\pm0.09$ \citep{Kunder2017} to [Fe/H] = $0.38\pm0.07$ \citep{Buder2021}.

An independent photometric estimate of the metallicity can be made based on the star's position on a color-magnitude diagram. Using the $V-K_s$ vs. $M_{Ks}$ calibration from \citet{Johnson2009} and \cite{Schlaufman2010}, we find that the color-mag position for TOI-1075\ ($V-K_s$ = 3.584, $M_{Ks}$ = 5.176) is only 0.056 mag below the mean main sequence for late K- and M-dwarfs, translating to photometric metallicity estimates of [Fe/H] = -0.08 \citep{Johnson2009} and -0.21 \citep{Schlaufman2010}. A similar estimate can be done using 2MASS and Gaia photometry by interpolating the photometry and metallicities of nearby M-dwarfs in \citet{Mann:2015}. As with $V-K_s$ vs. $M_{Ks}$, the star sits slightly below the Solar-metallicity sequence in $M_G$ using $B_P-R_P$, $J-K$, or $G-K_S$. This interpolation method gave us a metallicity estimate of [Fe/H] = -0.10$\pm$0.12, consistent with the first estimate from \citet{Johnson2009}.

We also analyzed the iodine-free PFS template spectrum using the publicly available code \texttt{SpecMatch-Emp} \citep{Yee2017}. This code matches a target spectrum with a library of observed spectra from stars with empirically determined stellar properties, and is particularly well-suited for the analysis of late-type stars. We recovered $T_{\rm eff} = 3824 \pm 70$~K, $R_\star = 0.57\pm0.06~R_\odot$, and $\mathrm{[Fe/H]} = -0.08 \pm 0.09$~dex. The \texttt{SpecMatch-Emp} recovered metallicity thus corroborates the photometric metallicity measurement.

\subsection{Stellar Variability}

WASP-South, an array of 8 cameras composed of Canon 200-mm, f/1.8 lenses backed by 2k$\times$2k CCDs, was the Southern station of the WASP transit-search survey \citep{2006PASP..118.1407P}. The field of TOI-1075\ was observed every year from 2008 to 2012, covering spans of 100 d to 180 d in each year. Within each night the cadence was typically 15-min, accumulating a total of 38\,000 data points.  TOI-1075\ is the only bright star in the 48\arcsec\, photometric extraction aperture. We searched the WASP data for a rotational modulation using the methods described in \citet{2011PASP..123..547M} but we find no significant modulation for any period between 1 and 100 days. Within each season the 95\%-confidence upper limit on the amplitude is 3~mmag. Combining all the years of data results in an upper limit of 1.6~mmag. The periodograms of each season of WASP-South data showing no significant rotation modulation are shown in Figure \ref{fig:wasp}. The peaks near 30~days are compatible with the residual effects of moonlight propagating through the pipeline at a low level, and thus are unlikely to be caused by TOI-1075.
The lack of any rotational modulation is unusual for a cool star.  \citet{2014ApJS..211...24M} report that 83\%\ of stars cooler than 4000~K in the Kepler field show a rotational modulation, with most having amplitudes in the range 3--10~mmag.  Hence TOI-1075\ is among the least photometrically variable $\sim$\,20\%\ of stars of its spectral type. TOI-1075's low levels of stellar variability support the finding that the star is at least 2~Gyr old.

\begin{figure}[!ht]
    \centering
    \includegraphics[width=\linewidth]{figures/toi1075_wasp.pdf}
\caption{Periodograms of each season of WASP data for TOI-1075, showing the absence of any significant rotational modulation.  The horizontal lines mark the 95\%-confidence false-alarm level.  The top panel is the periodogram for the 5 years of data combined. \label{fig:wasp}}
\end{figure}

\section{Exoplanet Detection \& Follow Up \label{sec:planetdata}}

\subsection{\emph{TESS}\ Time Series Photometry}

The \emph{TESS}\ primary mission surveyed the northern and southern ecliptic hemispheres in sectors measuring 24$^{\circ} \times$ 96$^{\circ}$, with near-continuous photometric coverage over $\sim$27~days. The TESS Primary Mission ran for 2 years (July~2018 -- July~2020), and consisted of 26~sectors. \emph{TESS}\ began its first extended mission in July 2020 which will end in September 2022 (when the second extended mission is scheduled to commence), and consists of 29~sectors.
TOI-1075\ (TIC~351601843, 2MASS~J20395334-6526579) was selected for transit detection observations by \emph{TESS}\ with 2-minute cadence as part of the Candidate Target List (CTL) -- a pre-selected target list prioritized for the detection of small planets \citep{stassun2018tess, Stassun2019}.
TOI-1075\ was observed by \emph{TESS}\ in Sector~13 from UT 2019 June 19 through 2019 July 18 during the primary mission, and again from UT 2020 July 4 through 2020 July 30 in Sector 27 during the first \emph{TESS}\ Extended Mission. The star fell on Camera~2 in both sectors.

The raw \emph{TESS}\ data for TOI-1075\ were processed with the Science Processing Operations Center Pipeline (SPOC; \citealp{Jenkins2016}) -- which performs pixel calibration, light curve extraction, de-blending from nearby stars, and removal of common-mode systematic errors -- and are available at the Mikulski Archive for Space Telescopes (MAST) website\footnote{https://mast.stsci.edu}. The SPOC data include both simple aperture photometry (SAP) flux measurements \citep{Twicken2010, Morris2017} and presearch data conditioned simple aperture photometry (PDCSAP) flux measurements \citep{Smith2012, Stumpe2012, Stumpe2014}. The instrumental variations present in the SAP flux are removed in the PDCSAP flux data.

We further detrended the \emph{TESS}\ PDCSAP data by median normalizing, flattening, fitting a low order spline, and removing 3-$\sigma$ outliers from each sector's flux measurements separately, before stitching the light curves together. The detrended \emph{TESS}\ PDSCAP data for Sector~13 and Sector~27, as well as the phase-folded light curve are shown in Figure \ref{fig:tessdataphased}.

\begin{figure*}[htb!]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{figures/toi1075_tess_data_phased_residuals_zoom_transits_indc.png}
\caption{\emph{TESS}\ light curves of TOI-1075. \textit{Top:} Detrended, normalized and flattened PDCSAP flux measurements for Sector~13 (left) and Sector~27 (right). Lighter gray points are the \emph{TESS}\ 2-minute cadence flux measurements; darker points are the same data binned into 30-minute intervals. The transits of TOI-1075~b\ are marked by orange triangles. \textit{Middle}: Phase-folded light curve on planet's orbital period (0.605~days), along with the best-fit transit model (red). Gray points are 2-minute cadence measurements; white points are the same data binned into 5-minute intervals. \textit{Bottom:} Residuals after the data have been subtracted from the best-fit model. \label{fig:tessdataphased}}
\end{figure*}

\subsubsection{\emph{TESS}\ Transit Detection}
The SPOC Transiting Planet Search (TPS; \citealp{jenkins2002impact, jenkins2010transiting}) pipeline searches for threshold crossing events (TCEs) in the PDCSAP light curve, applying an adaptive noise-compensating matched filter to account for stellar variability and residual observation noise. TCEs with a period of 0.605~days were detected independently in the SPOC transit search of the Sector~13 light curve\footnote{The gap at the beginning of the TESS Sector~13 data in Figure \ref{fig:tessdataphased} is a result of cadences being excluded from TPS due to the effects of rapidly changing scattered light and glints from the Earth and Moon.}, Sector~27 light curve, and multisector light curves from Sectors 13 and 27.

In order to rule out false positives that can mimic the planetary transit signal, we evaluated the star's data validation reports (DVR; \citealp{twicken2018kepler, li2019kepler}), which are generated from the SPOC 2-minute cadence data. The multisector DVR shows no evidence of secondary eclipses, odd/even transit depth inconsistencies, or correlations between the depth of the transit and the size of the aperture used to extract the light curve -- which would indicate that the transit signal originated from a nearby eclipsing binary. Additionally, the location of the transit source as shown in the DVR is consistent with the position of the target star -- the difference image centroiding test located the source of the transits due to TOI-1075~b\ to within $1.64\pm4.69$" of TOI-1075, which complements the speckle imaging observations. Upon passing these vetting checks, the transit signal was assigned the identifier TOI-1075.01 and announced by the \emph{TESS}\ TOI team. \citep{guerrero2021tess}.

\subsection{Ground-based Time-Series Photometry}

After it was alerted as a TOI, we acquired ground-based time-series follow-up photometry of TOI-1075\ during future times of transit predicted by the \emph{TESS}\ data. We used the {\tt TESS Transit Finder}, which is a customized version of the {\tt Tapir} software package \citep{Jensen:2013}, to schedule our transit observations.

\begin{figure*}[htb!]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{figures/TESS_LCO_MEARTH_lightcurves_jointfit_rvtrend_residuals_newbins.png}
\caption{Results of the \texttt{juliet} joint fit to the \emph{TESS}\ photometry, and ground-based LCOGT and MEarth-South photometry. \textit{Top:} Transit observations of the respective instruments phase-folded to the period of TOI-1075~b. The black curve is the best-fit \texttt{juliet} transit model, and the 68\% confidence interval is represented by the gray shaded region. The binned data points with errorbars are shown for clarity (white circles). The \emph{TESS}\ data are binned in 2-minute intervals, and the LCOGT and MEarth-South data are binned in 5-minute intervals. \textit{Bottom:} Residuals after the data have been subtracted from the best-fit model. \label{fig:transitsjointfit}}
\end{figure*}

\subsubsection{LCOGT 1\,m Observations}

We observed five full transits of TOI-1075\ in Pan-STARRS $z$-short band from the Las Cumbres Observatory Global Telescope \citep[LCOGT;][]{Brown:2013} 1.0\,m network on UTC 2019 August 25, 2019 August 26, 2019 September 23, 2019 September 24, and 2019 September 26 (Figure \ref{fig:transitsjointfit}). The first, third and fifth observations were conducted from the South African Astronomical Observatory (SAAO) node, and the second and fourth observations from the Siding Spring Observatory(SSO) node. The 1\,m telescopes are equipped with $4096\times4096$ SINISTRO cameras having an image scale of $0\farcs389$ per pixel, resulting in a $26\arcmin\times26\arcmin$ field of view. The images were calibrated by the standard LCOGT {\tt BANZAI} pipeline \citep{McCully:2018}. Photometric data were extracted using {\tt AstroImageJ} \citep{Collins:2017} and circular photometric apertures with radii in the range $6\farcs2$ to $7\farcs8$. The target star apertures exclude flux from all known nearby Gaia~DR3 and TESS Input Catalog stars. We detect the event on-target in all five data sets, which are included in the joint model of the system in this work.

\subsubsection{MEarth-South 0.4m Observations}

We observed two full transits of TOI-1075~b\ using the MEarth-South telescope array \citep{nutzman2008design, irwin2015mearth} at Cerro Tololo Inter-American Observatory (CTIO), Chile on UT 2019 September 22 and 2019 September 28 (Figure \ref{fig:transitsjointfit}). Observations were gathered for approximately 5.5 hours centered on the predicted time of mid-transit. The data were reduced using the standard MEarth processing pipeline (e.g. \citealt{berta2012transit}) with a photometric extraction aperture of $r = 14$ pixels (11.8\arcsec). Twelve light curves were observed across 6 telescopes, and were collected with an RG715 filter. All of the light curves contain meridian flips prior to the predicted time of ingress.  These were accounted for in the analysis of the light curves by allowing for separate magnitude zero points for each combination of telescope and side of the meridian to remove any residual flat fielding error. Some residuals in the out of transit baseline, likely due to color-dependent atmospheric extinction, were found so the final model also included a linear decorrelation against airmass.

\subsubsection{Previous Validation of TOI-1075~b}
Additionally, planet candidate TOI-1075.01 was statistically validated as a planet using \emph{TESS}\ and ground-based photometry in \citet{giacalone2022validation}. TOI-1075.01 was vetted with \texttt{DAVE} \citep{kostov2019discovery} which uses centroid offset analyses to identify evidence of false positives due to contamination from nearby stars, and with \texttt{TRICERATOPS} \citep{giacalone2020triceratops,giacalone2021vetting}, which calculates the Bayesian probability that the candidate is an astrophysical false positive. TOI-1075.01 showed no strong indicators of being a false positive in the aforementioned analysis, and was then validated as TOI-1075~b\ \citep{giacalone2022validation}.

\subsection{Time Series Radial Velocities}

We collected 18 precision radial velocity epochs of TOI-1075\ using the Planet Finder Spectrograph \citep[PFS;][]{crane2006,crane2008,crane2010} on the 6.5~m Magellan~II (Clay) telescope at Las Campanas Observatory in Chile. PFS is a slit-fed spectrograph that is wavelength calibrated using an iodine cell, and covers the wavelength range $391-734$~nm, though only the 500-620~nm range is used when measuring RV shifts. All PFS spectra are reduced and RVs extracted using a custom {\texttt{IDL}} pipeline based on \citet{butler1996} that regularly delivers sub-1 m~s$^{-1}$ precision. Each of the 18 PFS TOI-1075\ iodine observations consist of 3$\times$20-minute exposures, and were mostly obtained on a night-by-night basis between UT 2021 May 22 and UT 2021 November 15, although in some cases two observations were taken on the same night but separated in time by at least 1-2 hours. A two-hour, iodine-free template observation was obtained on UT 2021 May 29. All observations were taken with the default 0.3$\arcsec$ slit but in 3$\times$3 binning mode, resulting in a resolving power of R$\sim$110,000. The resulting unbinned (20-minute integration time) RVs have typical precisions of 2.5--3.0 m~s$^{-1}$, and are listed in full in Table \ref{tab:RVs}.

A generalized Lomb–Scargle (GLS) periodogram of the PFS RV data shows a significant peak at 0.605~days (Figure \ref{fig:PFSperiodogram}), which matches the orbital period of the planet candidate determined from the \emph{TESS}\ data. An additional significant peak at $\sim$15~days likely corresponds to a stellar activity signal. The stellar activity signal is well-separated from the planet period, and there are no stellar periodicity signals more dominant than the planet signal, nor any significant stellar signals that appear around the planet period. Additionally, we modeled the suggested stellar activity signal using a Gaussian Process (GP) and find no evidence to incorporate it into our final model (see Section \ref{subsubsec:rvmodelcomp}). Therefore, incorporating GPs for non-white noise models, or adding another term to our RV model fit was deemed unnecessary.

\begin{center}
\begin{longtable}{ccc}
\caption{PFS RV data of TOI-1075}\\
\hline Date (BJD$_{\rm TDB}$) & RV ($\ensuremath{\rm m\,s^{-1}}$) & $\sigma_{RV}$ ($\ensuremath{\rm m\,s^{-1}}$)\\
\hline \vspace{2pt}
2459356.868 & -12.48 & 2.90 \\
2459356.884 & -5.77 & 3.04 \\
2459356.897 & -14.11 & 2.65 \\
2459357.856 & -25.83 & 3.00 \\
2459357.870 & -8.09 & 2.52 \\
2459357.885 & -18.11 & 3.38 \\
2459358.848 & -30.56 & 2.40 \\
2459358.862 & -26.80 & 2.18 \\
2459358.876 & -27.47 & 2.30 \\
2459452.564 & -11.29 & 2.34 \\
2459452.578 & -11.31 & 2.49 \\
2459452.593 & -6.06 & 2.57 \\
2459452.665 & -12.61 & 2.69 \\
2459452.679 & -5.05 & 2.56 \\
2459452.693 & -6.42 & 3.01 \\
2459470.617	& 10.02	& 3.18 \\
2459470.631 & -6.18	& 3.05 \\
2459470.645 & 1.55 & 3.65 \\
2459471.561 & 10.02 & 2.61 \\
2459471.576 & 14.14 & 2.58 \\
2459471.590 & 13.93 & 2.55 \\
2459471.659 & 15.38 & 2.71 \\
2459471.673 & 13.14 & 2.52 \\
2459471.687 & 6.22 & 2.58 \\
2459473.537 & 11.41 & 2.48 \\
2459473.550 & 17.32 & 2.52 \\
2459473.565 & 20.98 & 3.25 \\
2459473.647 & 6.82 & 3.02 \\
2459473.661 & 10.98 & 3.19 \\
2459473.675 & 14.55 & 3.39 \\
2459474.554 & -3.83 & 2.69 \\
2459474.568 & 4.77 & 2.62 \\
2459474.583 & -2.40 & 3.05 \\
2459474.616 & 6.56 & 3.00 \\
2459474.630 & 2.55 & 3.02 \\
2459475.526 & -5.85 & 3.07 \\
2459475.540 & -3.90 & 2.47 \\
2459475.554 & -6.89 & 2.60 \\
2459501.537 & 7.60 & 2.90 \\
2459501.551 & -2.56 & 2.80 \\
2459501.565 & 4.48 & 2.86 \\
2459504.528 & 0.00 & 2.47 \\
2459504.542 & -9.94 & 2.75 \\
2459504.556 & -12.43 & 2.91 \\
2459505.549 & 11.69 & 4.34 \\
2459505.563 & 7.41 & 3.99 \\
2459505.578 & 4.74 & 3.25 \\
2459506.518 & -19.93 & 2.41 \\
2459506.532 & -1.76 & 2.45 \\
2459506.546 & -14.89 & 2.33 \\
2459531.523 & 7.30 & 2.49 \\
2459531.537 & 10.63 & 2.55 \\
2459533.536 & 7.42 & 2.67 \\
2459533.546 & 11.35 & 4.27 \\
\hline
\label{tab:RVs}
\end{longtable}
\end{center}

\begin{figure}[htb!]
    \centering
    \includegraphics[width=\linewidth,keepaspectratio]{figures/TOI1075_RV_LinearFitResiduals_Periodogram_9x6_new.png}
\caption{Generalized Lomb–Scargle periodogram of the radial velocity measurements from PFS. There is a clear peak at the period of TOI-1075~b\ (gray bar).  We identify a stellar activity signal at $\sim$15~days (red dashed line) which is well-separated from TOI-1075~b's orbital period. \label{fig:PFSperiodogram}}
\end{figure}

\section{System Parameters from \texttt{\lowercase{juliet}}}
\label{sec:juliet_jointfit}

To obtain precise parameters of the TOI-1075\ system, we performed a joint analysis of the \emph{TESS}, LCOGT, and MEarth-South photometry, and the PFS RV data using \texttt{juliet} \citep{espinoza2019juliet}. \texttt{juliet} is a fitting tool that uses nested sampling algorithms to efficiently sample a given parameter space, and allows for model comparison based on Bayesian evidences. \texttt{juliet} combines the publicly available packages for transits and RV modeling, \texttt{batman} \citep{kreidberg2015batman} and \texttt{radvel} \citep{fulton2018radvel}, respectively. We opted to implement \texttt{dynesty} \citep{speagle2020dynesty} as the nested sampling algorithm for our joint fitting, though a range of nested sampling algorithms are available to choose from.

\subsection{Transit Modeling}
For the transit modeling, \texttt{juliet} employs the \texttt{batman} package. We adopt a quadratic model to describe the limb darkening effect in the \emph{TESS}, LCOGT, and MEarth-South photometry, and parameterize it by employing the efficient, uninformative sampling scheme of \citet{kipping2013efficient}, and a quadratic law. We used a fixed dilution factor of 1 for all instruments, but considered free individual instrumental offsets. Instrumental jitter terms were taken into account and added in quadrature to the nominal instrumental errorbars.
We used uniform priors per the \citet{espinoza2018efficient} parameterization to explore the full physically plausible parameter space for the planet-to-star radius ratio, $p=R_{p}/R_{\star}$, and impact parameter, $b$.
Additionally, we defined a log-uniform prior on the stellar density, and then recovered the scaled semi-major axis ($a/R_*$) using Kepler's third law.

\subsection{RV Modeling}
The model that we selected for our RV joint fit analysis was composed of a circular Keplerian orbit for the transiting planet (ultra-short period planets are expected to be tidally circularized), and an additional linear long-term trend to constrain the non-Keplarian long-period signal present in the PFS RV data, whose period is longer than the current observation baseline. We assumed uniform wide priors for the systemic velocity, jitter term, and RV semi-amplitude of the PFS RVs, as well as the linear long-term trend parametrized by an intercept $B$, and a slope $A$. We measured a radial velocity semi-amplitude of $K = 10.95^{+1.50}_{-1.43}~\ensuremath{\rm m\,s^{-1}}$ for TOI-1075~b.

\subsubsection{RV Model Comparison} \label{subsubsec:rvmodelcomp}

\texttt{juliet} searches for the global posterior maximum based on the evaluation of the Bayesian log-evidence ($\ln\mathcal{Z}$), allowing us to perform model comparisons given the differences in $\Delta\ln\mathcal{Z}$. We modeled the PFS RV data with and without fitting a linear long-term trend to the data. The model with the linear long-term trend had a log-evidence of $\ln\mathcal{Z}$ = $-$204.01 $\pm$ 0.38, and the model without the linear long-term trend (a circular Keplarian model only) had a log-evidence of $\ln\mathcal{Z}$ = $-$212.95 $\pm$ 0.28, resulting in a $\Delta\ln\mathcal{Z}$ = 8.94. We selected the model with the linear long-term trend component following criteria described in \citet{trotta2008bayes}, which considers a $\Delta\ln\mathcal{Z}$ $>$ 2 as weak evidence that one model is preferred over the other, and a $\Delta\ln\mathcal{Z}$ $>$ 5 as strong evidence that one model is significantly preferred over the other, hence the additional model parameters are necessary to account for the long-period signal in the PFS RV data. We also considered a model composed of a circular Keplerian orbit, and a long-term trend parametrized by an intercept $B$, a slope $A$, and a quadractic/curvature coefficient, $Q$. The change in log-evidence was $\Delta\ln\mathcal{Z}$ $<$ 2 between the long-term trend parametrization with and without the additional parameter, $Q$, confirming that the RV data are legitimately fitted by a circular Keplerian model and a linear long-term trend. As an additional test, we added a stellar signal to the RV model using a GP to determine the effect of the 15-day period signal identified in the PFS GLS periodogram (Figure \ref{fig:PFSperiodogram}). We compared the models with and without the additional stellar signal and found no evidence ($\Delta\ln\mathcal{Z}$ $<$ 2) that the final model required the addition of a stellar signal.

We show the final transit and RV models of the joint fit based on the posterior sampling in Figures \ref{fig:transitsjointfit} and \ref{fig:RVsjointfit} respectively, the posterior parameters of our joint fit in Table \ref{tab:julietposteriors}, the selected priors for our joint fit in Table \ref{tab:julietpriors}, the obtained posterior probabilities in Figure \ref{fig:cornerposteriors}, and the derived planetary parameters of TOI-1075~b\ based on the posteriors of the joint fit in Table \ref{tab:julietderived}.\\

To summarize, the TOI-1075\ system consists of a late K-/early M-dwarf host star with at least one hot super-Earth planet, TOI-1075~b\ (see Table \ref{tab:julietderived}), which has a mass of  $M_{p}$ = 9.95$^{+1.36}_{-1.30}$~\mearth\ and radius of $R_{p}$ = \plradunc\, on a circular orbit with a period of 0.605~days. We derived a bulk density of $\rho$ = \plrhounc\ and an equilibrium temperature, assuming a zero albedo, of $T_{eq} = 1323\pm44$~K.

\begin{figure*}[htb!]
    \centering
    \includegraphics[width=\textwidth,keepaspectratio]{figures/PFS_full_RVs_trend_phase_residual_phasebin.png}
\caption{Results of the \texttt{juliet} joint fit to the PFS radial velocities. \textit{Top:} PFS RVs over time (black points), and full best-fit \texttt{juliet} model (gray curve). \textit{Middle:} Phase-folded RV measurements from PFS. The black curve is the best-fit \texttt{juliet} RV model, the gray points are the unbinned RVs, and the red diamonds are the RV measurements binned in 0.05 units of orbital phase. Errorbars are the quadrature sum of the PFS internal uncertainties, and the RV jitter estimate from the \texttt{juliet} fit.The best-fit RV semi-amplitude is $K = 10.95^{+1.50}_{-1.43}~\ensuremath{\rm m\,s^{-1}}$. \textit{Bottom:} RV residuals after the data have been subtracted from the best-fit model. \label{fig:RVsjointfit}}
\end{figure*}

\section{Discussion}
\label{sec:discussion}

Our RV measurements of TOI-1075~b\ constrain the planetary mass with an uncertainty of $\sim14$\%, and the \emph{TESS}, LCOGT and MEarth-South light curves constrain the planetary radius with an uncertainty of $\sim7$\%. Thus, TOI-1075~b\ belongs to the small group of super-Earths with precisely measured masses\footnote{Only planet masses measured by the radial velocity method are considered, in order to avoid differences in the planet mass distribution with other methods.} and radii, or those planets with measurement precision better than 30\% (see Figure \ref{fig:MR}). With precise mass and radius measurements in hand for TOI-1075~b, we discuss implications for radius valley studies, the potential for planetary atmospheric characterization, potential planetary compositions, planetary bulk density, and possible planet formation mechanisms. We also discuss constraints on a potential second planet in the TOI-1075\ system.

\subsection{Implications for the Radius Valley around M-dwarfs}

Several physical mechanisms/theoretical models have been proposed to explain the existence of the radius gap. These mechanisms include  thermally driven atmospheric mass loss, e.g., photoevaporation \citep{lopez2012thermal, chen2016evolutionary, owen2017evaporation} and core-powered mass loss \citep{ginzburg2018core,gupta2019sculpting,gupta2020signatures}; and gas-poor formation -- a natural outcome of planet formation, where rocky super-Earths are a result of formation in gas-poor environments, without requiring any atmospheric escape \citep{lee2014make,lee2016breeding,lopez2018formation,lee2021primordial}. The slope of the radius valley (in period-radius space) can be used to discern between a thermally driven mass-loss model or a gas-poor formation model \citep{lopez2018formation,gupta2020signatures,lee2021primordial}.

The slope of the radius valley around Sun-like stars has been characterized using data from the \textit{Kepler} and \textit{K2} missions, and both thermally driven mass-loss and gas-poor formation models are favored in this stellar-mass regime \citep{van2018asteroseismic,martinez2019spectroscopic}. However, around low-mass mid-K- to mid-M-dwarfs, \citet{cloutier2020evolution} found tentative evidence that the slope of the radius valley is consistent with predictions from gas-poor formation. Additionally, the radius gap decreases as stellar radius decreases, and the radius gap is centered at $1.54\pm0.16$~R$_{\oplus }$ for low-mass K- and M-dwarf stars (vs.~1.75~R$_{\oplus }$ for Sun-like stars in \citealt{fulton2017california}).

TOI-1075~b\ lies between the predicted slopes of the thermally driven mass loss model and gas-poor formation model, and hence within the M-dwarf radius valley created by these mechanisms (the M-dwarf radius valley ranges from 1.5-2.5~R$_{\oplus }$ between the predicted slopes -- see Figure 9 in \citealp{cloutier2021toi1634}). TOI-1075~b's orbital period (0.605~days) and size (\plradunc) make it a ``keystone planet" -- a valuable target to conduct tests of competing radius valley models across a range of stellar masses, using precise planetary mass and radius measurements \citep{cloutier2020evolution}. TOI-1075~b\ joins TOI-1235~b \citep{cloutier2020toi1235, bluhm2020precise}, TOI-776~b \citep{luque2021planetary}, TOI-1685~b \citep{bluhm2021ultra}, and TOI-1634~b \citep{cloutier2021toi1634} as keystone planets that will help elucidate the physical mechanism that formed the radius valley around early M-dwarfs. Distinguishing between the two atmospheric loss mechanisms will require the discovery of additional keystone planets for statistical studies and population analysis, as well as atmospheric studies of TOI-1075~b\ and other keystone planets to provide observational evidence to validate model predictions.

\subsection{Atmospheric Characterization Prospects \label{subsec:atmchr}}

Super-Earths with $R_{p} >$ 1.6~R$_{\earth}$ are expected to have a substantial H/He atmosphere \citep{rogers2015most}, and though TOI-1075~b's radius (\plradunc) places it just above the radius gap, its bulk density is inconsistent with the presence of a low mean-molecular weight envelope. Based on TOI-1075~b's predicted composition (see Section \ref{subsec:plcompdens}) and ultra-short orbital period, we do not expect the planet to have retained a H/He envelope. But, TOI-1075~b\ could either have: no atmosphere (bare rock); a metal/silicate vapor atmosphere with a composition set by the vaporizing magma-ocean on the surface \citep{schaefer2009chemistry,ito2015theoretical} since TOI-1075~b's equilibrium temperature is hot enough to melt a rocky surface \citep{mansfield2019identifying}; or, especially at the low-end of its allowed mean density range, possibly a thin H/He or CO$_2$ or other atmosphere. A detailed atmospheric model is needed to determine possible atmospheric compositions for TOI-1075~b, which is beyond the scope of this work.

We calculated the emission spectroscopy metric (ESM) and transmission spectroscopy metric (TSM), as defined by \citet{kempton2018framework}, to determine TOI-1075~b's potential for atmospheric characterization. Using the stellar parameters reported in Section \ref{sec:SED}, and the planetary parameters in Table \ref{tab:julietderived}, we obtain ESM~$= 10.1 \pm 1.6$ and TSM~$= 29 \pm 8$.

TOI-1075~b\ is a good candidate for emission spectroscopy with the James Webb Space Telescope (\emph{JWST}). The planet may have a mineral-rich atmosphere consisting of metal and silicate vapors, since its equilibrium temperature is high enough to melt silicate rock \citep{schaefer2009chemistry, leger2011extreme, ito2015theoretical, ito2021hydrodynamic}. If TOI-1075~b\ has no atmosphere, its surface may be characterized via secondary eclipse observations. With an ESM value of $10.1 \pm 1.6$, TOI-1075~b\ is well above the threshold ESM of 7.5 suggested by \citet{kempton2018framework} for a high-quality atmospheric characterization target. TOI-1075~b\ is one of only 8 super-Earths: 55~Cancri~e \citep{bourrier201855cnc}, HD~213885~b \citep{espinoza2020hd}, HD~3167~b \citep{christiansen2017three}, TOI-431~b \citep{osborn2021toi}, TOI-500~b \citep{serrano2022low}, TOI-1807~b \citep{nardiello2022gapstoi1807} and K2-141~b \citep{malavolta2018ultra}, with mass measurement precision $>5\sigma$ and with V/J/H/K $<$ 13 mag, that has an ESM $>$ 7.5. It is also the only super-Earth above the radius gap in the temperature range 1250--1750 K\footnote{https://tess.mit.edu/science/tess-acwg/ (as of 10/05/2022)}, which will allow us to probe an intermediate temperature range of hot super-Earths and explore the atmospheric species that have volatilized in this regime.

In addition to having an ESM value above the suggested threshold, TOI-1075~b\ can also be observed by \emph{JWST}\ for $\sim200$~days per year. Thus, TOI-1075~b\ is accessible to \emph{JWST}\ for a significant portion of the year, which allows for more flexibility when planning observations.

\citet{kempton2018framework} suggests that TSM~$> 90$ be used as the threshold for planets with 1.5~R$_{\oplus }$ $<$ R$_{p}$ $<$ 10~R$_{\oplus }$. TOI-1075~b\ does not meet this criteria because, due to its high mass and hence high surface gravity, it is unlikely to have an extended atmosphere that can be probed in transmission.

\subsection{Planetary Composition and Density \label{subsec:plcompdens}}

The measured mass and radius of TOI-1075~b\ result in a planetary bulk density of \plrhounc, which is almost twice as dense as the Earth ($\rho_{\earth}$ = 5.51~g cm$^{-3}$). Comparing TOI-1075~b\ with the theoretical composition models of \citet{zeng_new_2021} and Lin et al. (in prep), the planet's bulk density is consistent with a 35\% Fe + 65\% silicates by mass composition (see Figure \ref{fig:MR}).

To simulate TOI-1075~b's interior, we numerically integrate three equations -- the mass of a spherical shell, hydrostatic equilibrium, and the equation of state (EOS) -- from the planet's center to the surface with a step size of 100~m using a planetary interior simulation code (Lin et al. (in prep)). The code is validated against recent mass-radius curves calculated by \cite{zeng_new_2021}. We assume a completely differentiated planet with an iron core and a mantle consisting of silicates. For iron and silicates, we adopt a second-order adapted polynomial EOS developed by \cite{holzapfel_coherent_2018}, using EOS coefficients listed in \cite{zeng_new_2021}. The inner boundary conditions of the simulated planets are assumed to be $M(0)=0$ and $P(0)=P_c$, where $P_c$ is the central pressure. We switch from the iron core to the silicate mantle when the desired core mass has been reached. The iteration terminates when the outer boundary condition $P(R)\le1$ bar is satisfied. We use a bisection method to search for the $P_c$ that produces $M(R)=M_p$ for a given core-mass fraction (CMF). Using the mean measured mass and radius of TOI-1075~b, we compute a mean CMF of $0.35$. We then calculate a core-radius fraction (CRF) by dividing the radius of the iron layer by the total radius of the planet, resulting in a mean CRF of $0.52$, or $0.93\,$R$_{\earth}$.

We further consider the most and least dense scenarios permissible by the mass and radius error bars. In the most dense scenario ($M_p = 11.31\,$M$_{\earth}$, $R_p = 1.709\,$R$_{\earth}$), we compute a CMF of 0.61 and a CRF of 0.67, or $1.15\,$R$_{\earth}$. In the least dense scenario ($M_p = 8.65\,$M$_{\earth}$, $R_p = 1.91\,$R$_{\earth}$), even a coreless pure silicates composition (CMF = 0, CRF = 0) cannot explain the low-end density of the planet. Even though such a coreless planet is unphysical from a planet formation perspective, we include this extreme scenario for completeness.

Though we have precisely measured the mass of TOI-1075~b\ to $>7\sigma$ (9.95$^{+1.36}_{-1.30}$~\mearth), the uncertainty on the mass measurement leads to a wide range of possible CMF ($0.35^{+0.26}_{-0.35}$) and CRF ($0.52^{+0.15}_{-0.52}$) values, which can only be resolved with more precise mass and radius measurements not currently available. In the following sections, we discuss possible formation scenarios that could result in the mean density, most dense, and least dense scenarios.

\begin{figure*}[htb!]
    \centering
    \includegraphics[width=\textwidth, keepaspectratio]{figures/MR_composition_density_curves_combined_v8_review.png}
\caption{Mass–radius diagram for small planets ($R_p$ $<$ 2~R$_{\earth}$ and $M_p$ $<$ 10~M$_{\earth}$) with measured mass and radius uncertainties below 30\%, as listed in the NASA Exoplanet Archive (only planet masses measured by the radial velocity method are considered). Planets are colored according to their calculated equilibrium temperature (assuming zero-albedo and efficient heat redistribution). Super-Earth planets have square symbols, and super-Mercury planets have bold diamond symbols. The gray shaded region denotes the radius gap for low-mass stars centered at $1.54\pm0.16$~R$_{\oplus }$ \citep{cloutier2020evolution}. The terrestrial solar system planets are plotted for reference. \textit{Left:} Mass-radius diagram with curves of constant composition. The solid lines are theoretical internal composition curves (\citealt{zeng_new_2021}; Lin et al. (in prep)) from top to bottom: 100\% H$_2$O (blue), 100\% silicates (green), a mixture of 30\% Fe and 70\%  silicates by mass (orange),  a mixture of 50\% Fe and 50\%  silicates by mass (light brown), a mixture of 70\% Fe and 30\%  silicates by mass (dark brown), and 100\% Fe (black). TOI-1075~b's bulk density is consistent with a 35\% Fe + 65\% silicates by mass composition. \textit{Right:} Mass-radius diagram with curves of constant density relative to Earth. TOI-1075~b's density is $\sim$1.75 times greater than the Earth. \label{fig:MR}}
\end{figure*}

\subsubsection{TOI-1075~b's Mean Density and Composition}

The mean mass and radius ($M_p = 9.95\,$M$_{\earth}$, $R_p = 1.791\,$R$_{\earth}$) of TOI-1075~b\ result in a bulk density of 9.32~g~cm$^{-3}$. While TOI-1075~b's mean bulk density is $\sim1.75\times$ that of Earth, its mean CMF (0.35) and CRF (0.52) are consistent with a predominantly rocky, Earth-like composition and internal structure. TOI-1075~b's uncompressed density is 4.91~g~cm$^{-3}$, which is similar to the uncompressed density of the Earth (4.79~g~cm$^{-3}$), further supporting our findings of an Earth-like composition for TOI-1075~b.

TOI-1075~b\ is the most massive and densest super-Earth with 1.6~R$_{\oplus }$ $<$ R$_{p}$ $<$ 2~R$_{\oplus }$ discovered to date. The similarity in TOI-1075~b's CMF relative to Earth (CMF$_{\earth}$ = 32.5\%, \citealp{seager2007mass}) supports our finding that TOI-1075~b\ likely lacks a massive low mean-molecular weight envelope which, if present, would have corresponded to a larger observed radius for a given mass.

A handful of massive terrestrial planets like TOI-1075~b\ have been discovered, but these objects are relatively rare. Planets similar in radius and Earth-like composition to TOI-1075~b\ are listed in Table \ref{tab:massiveplanets}.

\subsubsection{TOI-1075~b\ As A Super-Mercury}

In the most dense scenario ($M_p = 11.31\,$M$_{\earth}$, $R_p = 1.710\,$R$_{\earth}$, $\rho = 11.37$~g~cm$^{-3}$), we consider the possibility of TOI-1075~b\ as a super-Mercury. The term ``Super-Mercuries" generally refers to planets that are super-Earth sized with enhanced uncompressed bulk densities, as compared to Earth-like planets. The high density has been interpreted to be indicative of a high iron-mass fraction, analogous to the solar system planet Mercury \citep{marcus2010minimum,adibekyan2021compositional}, and the formation mechanism for these planets is still unknown. Based on TOI-1075~b's bulk density (\plrhounc), if the planet's mass is at the high range, with a high core-mass fraction in the allowed range (CMF = $0.35^{+0.26}_{-0.35}$), it could be a super-Mercury. The small group of recently identified super-Mercuries include: K2-38~b, K2-106~b, K2-229~b, Kepler-107~c, and Kepler-406~b \citep{adibekyan2021compositional}, as well as GJ~367~b \citep{lam2021gj} and HD~137496~b \citep{silva2022hd}, which are represented by bold data points in Figure \ref{fig:MR}.

We investigated possible formation mechanisms that could result in the high density, and CMF and CRF ranges observed for TOI-1075~b\ including: giant impacts \citep{marcus2010minimum, benz2008origin, asphaug2014mercury, liu2015giant, scora2020chemical, cambioni2021effect, cambioni2022metal}; in-situ formation \citep{weidenschilling1978iron, kruss2018seeding, johansen2022nucleation}; an initially metal-rich protoplanetary disk composition \citep{veyette2017physically, adibekyan2021compositional, schulze2021probability, souto2022detailed}; and the decompressed core of an evaporated gas giant \citep{hebrard2003evaporation, mocquet2014very}, but we do not have sufficient information to resolve the planet's formation mechanism.

\subsubsection{TOI-1075~b\ As A Low-density Planet}

In the least dense scenario ($M_p = 8.65\,$M$_{\earth}$, $R_p = 1.907\,$R$_{\earth}$, $\rho = 7.47$~g~cm$^{-3}$), our interior model results in a coreless silicate planet, and requires an additional water layer to account for the inflated radius. Such a coreless planet is unphysical from a planet formation perspective -- a planet with a mass of TOI-1075~b\ is expected to have fully differentiated e.g. \citet{rubie2015accretion,cambioni2021effect}. A water layer would be physically unstable at TOI-1075~b's equilibrium temperature. Therefore, layers less dense than silicates must be added to the model to fit the minimum mass and maximum radius scenario. Possible candidates for these layers include low-pressure silicate phases, or a metal/silicate vapor atmosphere.

\begin{table*}[htb!]
    \caption{Massive terrestrial planet parameters}
    \centering
    \begin{tabular}{lcccl}
    \hline
    Planet Name & Planet Radius ($R_{\earth}$) & Planet Mass ($M_{\earth}$) & T$_{eq}$ (K) & Reference \\
     \hline
    TOI-1075~b\ & $1.791_{-0.081}^{+0.116}$ & $9.95_{-1.30}^{+1.36}$ & 1323$\pm44$& This work\\
     Kepler-20~b & ${1.868}_{-0.034}^{+0.066}$ & ${9.70}_{-1.44}^{+1.41}$ & $1105\pm37$ & \citet{buchhave20161}\\
    LHS~1140~b & $1.727\pm0.032$ & $6.98\pm0.89$ & $235\pm5$ & \citet{ment2019second}\\
    TOI-1235~b & $1.738_{-0.076}^{+0.087}$ & ${6.91}_{-0.85}^{+0.75}$ & $754\pm18$ & \citet{cloutier2020toi1235}\\
    HD~213885~b & $1.74\pm0.05$ & $8.8\pm0.6$ & $2128\pm14$ & \citet{espinoza2020hd}\\
    WASP-47~e & $1.808\pm0.026$ & $6.77\pm0.57$ & $2514\pm70$ & \citet{bryant2022revisiting}\\
    K2-216~b & $1.72\pm0.06$ & $8.18\pm1.65$ & $1217\pm34$ & \citet{clark2022galah}\\
    \hline
    \label{tab:massiveplanets} \end{tabular}
\end{table*}

\subsection{A Potential Second Planet in the System \label{subsec:secondpl}}

We find a linear long-term trend in the PFS RV data (see Section 4.2) whose period is longer than the baseline of our RV observations. This may indicate the presence of a second planet in the system. We place lower limits on the period, semi-amplitude, and mass of a second planet candidate as the source of the long-term trend.

The orbital period of the second planet candidate is at least twice the observing baseline of the PFS RV observations (otherwise we would have seen the RV trend turn over before our observations concluded as the potential planet passed through its quadrature phase). The PFS observations were taken over a period of $176.68$~days (T$_{\rm baseline}$), hence the orbital period of the planet candidate should be at least $353$~days.

Taking the best-fit linear trend results from \texttt{juliet}, the RV data is shifted $0.130^{+0.017}_{-0.018}~\rm m\,s^{-1}\,day^{-1}$. Thus, the RV semi-amplitude of the planet candidate must be at least (T$_{\rm baseline}$ $\times$ RV Trend)/2 = $11.48~\ensuremath{\rm m\,s^{-1}}$.

Combining the minimum orbital period ($353$~days) and the minimum RV semi-amplitude ($11.48~\ensuremath{\rm m\,s^{-1}}$), and assuming a circular Keplarian orbit, the minimum mass of the second planet candidate is $\sim87~M_{\earth}$ or $\sim0.28~M_{\rm Jupiter}$.

The presence of a second planet candidate motivates us to collect additional RV data for this system in order to determine the period and measure the mass of the second planet candidate, while also improving the uncertainty on the mass measurement of TOI-1075~b. Additionally, there are currently a handful of systems consisting of a USP planet and a close-in companion, with periods ranging from a few days to tens of days e.g. K2-106 \citep{guenther2017k2}, K2-141 \citep{malavolta2018ultra}, K2-229 \citep{santerne2018earth}, TOI-500 \citep{serrano2022low}. These close-in companions could be responsible for migrating the USP planets to their current positions \citep{pu2019low, millholland2020formation, serrano2022low}. The possible existence of such a close-in companion in the TOI-1075\ system serves as additional motivation for further RV follow-up of the system.

Determining the source of the long-term trend, and more accurately measuring TOI-1075~b's planet mass and parameters will further detailed planet formation, planet migration and atmospheric characterization efforts, since a planet's gravity plays an important role in its collisional history and interpreting atmospheric spectra \citep{batalha2019precision}.

\section{Conclusions\label{sec:conclusions}}

We report the discovery and confirmation of TOI-1075~b, a transiting, ultra-short period, hot super-Earth orbiting a nearby ($d$ = 61.4 pc) late K-/early M-dwarf star. Using photometric observations from \emph{TESS}, LCOGT, and MEarth-South, and radial velocity observations from PFS, we precisely measure the radius and mass of TOI-1075~b\ to be \plradunc\ and 9.95$^{+1.36}_{-1.30}$~\mearth, respectively. Our PFS radial velocity data also suggest the presence of a second planet candidate in the system, with a minimum mass of $\sim87~M_{\earth}$ and a minimum orbital period of $\sim353$~days. TOI-1075~b\ has a bulk density of \plrhounc, consistent with a composition of 35\% iron by mass, and a core-radius fraction of 52\%. TOI-1075~b\ is a good candidate for emission spectroscopy with \emph{JWST}, which we can use to characterize a potentially mineral-rich atmosphere. \emph{TESS}\ is scheduled to observe TOI-1075\ again in Year~5, Sector~67 (July 2023), which will provide a more precise planet radius, and the ability to search for variations in the planet period on a 4-year timescale. TOI-1075~b\ is a massive, dense, high temperature, ultra-short period super-Earth inside of the M-dwarf radius valley, making the system ideal for testing planet formation and evolution theories, density enhancing mechanisms, and theoretical models related to atmospheric loss.

\clearpage
\startlongtable
\begin{deluxetable*}{llccc}
\tabletypesize{\small}
\tablewidth{\linewidth}
\tablecaption{Median values and 68\% confidence interval for posterior parameters from joint photometric and radial velocity \texttt{juliet} analysis for the TOI-1075 system.}
\tablehead{\colhead{Parameter} & \colhead{Units} & \colhead{Values}}
\startdata
\smallskip\\\multicolumn{2}{l}{Stellar Parameters:}&\smallskip\\
$\rho_*$ & Density (g cm$^{-3}$) & $3.73^{+0.99}_{-1.67}$\\
\smallskip\\\multicolumn{2}{l}{Planet Parameters:}&TOI-1075 b\smallskip\\
$P$ &Period (days) &$0.6047328\pm0.0000032$\\
$T_0$ & Time of transit center (\ensuremath{\rm {BJD_{TDB}}}) &$2458654.2510^{+ 0.00040}_{-0.00050}$\\
$r_1$ & Parametrization of \citet{espinoza2018efficient} for $b$ & $0.60^{+0.20}_{-0.17}$\\
$r_2$ & Parametrization of \citet{espinoza2018efficient} for $R_p/R_*$ & $0.0282^{+0.0019}_{-0.0013}$\\
$K$ & RV semi-amplitude (\ensuremath{\rm m\,s^{-1}}) & $10.95^{+1.50}_{-1.43}$\\
$e$ & Eccentricity (fixed) & $0.00$\\
\smallskip\\\multicolumn{2}{l}{Photometry Parameters:}\\
\multicolumn{2}{l}{}&MEarth-South&LCOGT&TESS\smallskip\\
$M$ & Relative flux offset & $0.0000$ & $0.0000$ & $-0.0001$\\
$\sigma$ &  Jitter term for light curve (ppm) & $901.29^{+86.80}_{-91.69}$ & $471.24^{+58.31}_{-62.17}$ & $3.21^{+25.84}_{-2.86}$\\
$q_{1}$ & Quadratic limb-darkening parametrization \citep{kipping2013efficient} & $0.54^{+0.28}_{-0.31}$ & $0.34^{+0.33}_{-0.23}$ & $0.59^{+0.24}_{-0.25}$\\
$q_{2}$ & Quadratic limb-darkening parametrization \citep{kipping2013efficient} & $0.47^{+0.33}_{-0.30}$ & $0.39^{+0.35}_{-0.27}$ & $0.47^{+0.29}_{-0.28}$\\
\smallskip\\\multicolumn{2}{l}{RV Parameters:}\smallskip\\
$\mu_{PFS}$ & Systemic velocity for PFS (\ensuremath{\rm m\,s^{-1}}) & $-67.3^{+24.7}_{-20.1}$ \\
$\sigma_{PFS}$ &  Jitter term for PFS (\ensuremath{\rm m\,s^{-1}}) & $6.60^{+0.81}_{-0.69}$ \\
$A$ & Slope of linear long-term RV trend (\ensuremath{\rm m\,s^{-1}}~day$^{-1}$) & $0.130^{+0.017}_{-0.018}$ \\
$B$ & Intercept of linear long-term RV trend (\ensuremath{\rm m\,s^{-1}}) & $-67.3^{+23.1}_{-20.6}$ \\
\label{tab:julietposteriors} \enddata
\end{deluxetable*}

\startlongtable
\begin{deluxetable*}{llccc}
\tabletypesize{\small}
\tablewidth{\linewidth}
\tablecaption{Median values and 68\% confidence interval for derived parameters from joint photometric and radial velocity \texttt{juliet} analysis for TOI-1075 b.}
\tablehead{\colhead{Parameter} & \colhead{Units} & \colhead{Values}}
\startdata
\smallskip\\\multicolumn{2}{l}{Derived Transit Parameters:}&\smallskip\\
$p=R_p/R_*$ & Radius of planet in stellar radii & $0.0282^{+0.0019}_{-0.0013}$\\
$a/R_*$ & Semi-major axis in stellar radii  & $4.40^{+0.89}_{-0.91}$\\
$b=(a/R_*)cos(i_p)$ & Transit impact parameter & $0.40^{+0.29}_{-0.25}$\\
$i_p$ & Inclination (degrees) & $84.67^{+3.34}_{-3.93}$\\
\smallskip\\\multicolumn{2}{l}{}&MEarth-South&LCOGT&TESS\smallskip\\
$u_{1}$ & Linear limb-darkening coefficient & $0.61^{+0.52}_{-0.40}$ & $0.39^{+0.46}_{-0.27}$ & $0.68\pm0.39$\\
$u_{2}$ & Quadratic limb-darkening coefficient & $0.035^{+0.043}_{-0.041}$ & $0.10^{+0.32}_{-0.37}$ & $0.042^{+0.044}_{-0.039}$\\
\smallskip\\\multicolumn{2}{l}{Derived Physical Parameters:}&TOI-1075 b\smallskip\\
$R_p$ & Radius (R$_{\earth}$) &$1.791^{+0.116}_{-0.081}$\\
$M_p$ & Mass (M$_{\earth}$) &$9.95^{+1.36}_{-1.30}$\\
$\rho_p$ & Density (g cm$^{-3}$) &$9.32^{+2.05}_{-1.85}$\\
$a$ & Semi-major axis (AU) &$0.01159^{+0.00023}_{-0.00020}$\\
$T_{eq}$ & Equilibrium temperature (K) & $1323\pm44$\\
$g_p$ & Surface gravity (m~s$^{-2}$) & $30.0^{+5.4}_{-4.8}$\\
$S_{p}$ &Insolation (S$_{\earth}$) & $509.2^{+18.7}_{-20.4}$\\
\label{tab:julietderived} \enddata
\end{deluxetable*}

\acknowledgements

This paper includes data collected by the \emph{TESS}\ mission. Funding for the \emph{TESS}\ mission is provided by NASA's Science Mission directorate.
This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile.
This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.
This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.
Some of the observations in the paper made use of the High-Resolution Imaging instrument Zorro obtained under Gemini LLP Proposal Number: GN/S-2021A-LP-105. Zorro was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Zorro was mounted on the Gemini North (and/or South) telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).
The MEarth Team gratefully acknowledges funding from the David and Lucile Packard Fellowship for Science and Engineering (awarded to D.C.). This material is based upon work supported by the National Science Foundation under grants AST-0807690, AST-1109468, AST-1004488 (Alan T. Waterman Award), and AST-1616624, and upon work supported by the National Aeronautics and Space Administration under Grant No. 80NSSC18K0476 issued through the XRP Program. This work is made possible by a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.
This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).
Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). The specific observations analyzed can be accessed via \dataset[https://doi.org/10.17909/t9-wpz1-8s54]{https://doi.org/10.17909/t9-wpz1-8s54}.
This research has made use of the NASA Exoplanet Archive \citep{planetarysystemsexoarchive}, made available by the NASA Exoplanet Science Institute at IPAC, which is operated by the California Institute of Technology under contract with the National Aeronautics and Space Administration.
This research has made use of NASA's Astrophysics Data System.
This research has made use of the Exoplanet Follow-up Observation Program (ExoFOP; \citealp{https://doi.org/10.26134/exofop5}) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program."
This research made use of Astropy, a community-developed core Python package for Astronomy \citep{Astropy2013}.
This publication makes use of VOSA, developed under the Spanish Virtual Observatory (https://svo.cab.inta-csic.es) project funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00.
VOSA has been partially updated by using funding from the European Union's Horizon 2020 Research and Innovation Programme, under Grant Agreement nº 776403 (EXOPLANETS-A).

\facilities{TESS, Magellan:Clay (Planet Finder Spectrograph), Gemini-South (Zorro), SOAR, LCOGT, MEarth}

\software{{AstroImageJ \citep{Collins:2017}, Astropy \citep{Astropy2013}, TAPIR \citep{Jensen:2013}, juliet \citep{espinoza2019juliet}, dynesty \citep{speagle2020dynesty}}}

\bibliographystyle{apj}

redpajama_set_name: RedPajamaArXiv


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