A Pair of TESS Planets Spanning the Radius Valley around the Nearby Mid-M Dwarf LTT 3780

From UT 2019 February 28 to March 26, LTT 3780 was observed in TESS sector 9 (i.e., orbits 25 and 26) for 27.26 days with CCD 1 on camera 1. As a member of the cool dwarf target list (Muirhead et al. 2018), LTT 3780 was included in the TIC and in the candidate target list (Stassun et al. 2018) such that its light curve was sampled at a 2 minute cadence. These data were processed by the NASA Ames Science Processing Operations Center (SPOC; Jenkins et al. 2016). The resulting Presearch Data Conditioning Simple Aperture Photometry (PDCSAP; Smith et al. 2012; Stumpe et al. 2012, 2014) light curve of LTT 3780 was corrected for dilution by known contaminating sources within the photometric aperture with a dilution factor of 0.713. According to the sector 9 data release notes,⁴⁶ the level of scattered light from the Earth in camera 1 CCD 1 at the start of each orbit was high and resulted in no photometry or centroid positions being calculated during the first 1.22 days of orbit 25 or the first 1.12 days of orbit 26. Data collection was also paused for 1.18 days for data downloading close to the spacecraft's time of perigee passage. Overall, a total of 24.08 days of science data collection were performed in TESS sector 9.

A sample image of the field surrounding LTT 3780 from the TESS target pixel files is shown in Figure 1. The TESS photometric aperture used to produce the PDCSAP light curve was selected to maximize the photometric signal-to-noise ratio (S/N; Bryson et al. 2010) and is overlaid in Figure 1. Blending in the TESS photometry by nearby sources is unsurprising given the large (21'') TESS pixels and the 1' FWHM of its point-spread function, coupled with the large number density of 37 sources within 25 (Gaia Collaboration et al. 2018; Lindegren et al. 2018). In Figure 1, the low-resolution TESS image is compared with an example ground-based image taken with the 1 m telescope at the Cerro Tololo Inter-American Observatory (CTIO) location of the Las Cumbres Observatory Global Telescope (LCOGT) network. The LCOGT z_s-band image features a pixel scale of 039, which is equivalent to a spatial resolution that is 54 times higher than in the TESS images. The LCOGT image clearly depicts the position of LTT 3780 within the TESS aperture and the positions of 24 nearby sources from the Gaia DR2. The relative positions of the neighboring sources to the TESS photometric aperture reveal how the aperture was optimized to minimize contamination by the nearby bright sources, including the binary companion star LP 729-55 at 161 east of LTT 3780's position.

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In the subsequent transit search conducted by the SPOC using the Transiting Planet Search (TPS) Pipeline Module (Jenkins 2002; Jenkins et al. 2010), two transiting planet candidate signals were flagged and subsequently passed a set of internal data validation tests (Twicken et al. 2018; Li et al. 2019). The planet candidates TOI-732.01 and 02 had reported periods of 0.768 and 12.254 days, corresponding to 28 and two observed transits, respectively. However, focusing solely on TESS measurements wherein the quality flag QUALITY equals zero, indicating the reliability of those measurements, the second transit of TOI-732.02 is only partially resolved, as its ingress is largely contaminated. Although the SPOC does not make an identical cut based on the QUALITY flag, the SPOC-reported orbital period of TOI-732.02 is found to be underestimated by about 3 minutes, as we will learn from our follow-up transit light-curve analysis (Section 3.3).

The initially reported depth for each planet candidate was 1253 ± 106 and 3417 ± 283 ppm, corresponding to preliminary planetary radii of 1.44 ± 0.07 and 2.38 ± 0.12 R_⊕ using the stellar radius reported in Table 1. Note that these planet parameters are preliminary and will be refined in our analysis of the TESS light curve in Section 4.1.

3.2.1. TRES Spectroscopy

TESS's large pixels (21'') result in significant blending of the LTT 3780 light curve with nearby sources, including its visual binary companion at 161 to the east (with a TESS magnitude difference ΔT = 2.42; see Figure 1). We obtained seeing-limited photometric follow-up observations of the LTT 3780 field close to the expected transit times of each planet candidate as part of the TESS Follow-up Observing Program (TFOP). The example image from this follow-up campaign in Figure 1 reveals how individual sources are resolved, which enabled the confirmation of the transit events on target and the scrutiny of nearby sources for nearby eclipsing binaries (EBs). Follow-up efforts were scheduled using the TESS Transit Finder, which is a customized version of the Tapir software package (Jensen 2013). Unless otherwise noted, the photometric data were extracted and detrended using the AstroImageJ software package (AIJ; Collins et al. 2017). The resulting light curves were detrended with any combination of time (i.e., a linear trend), airmass, and total background counts as necessary in attempts to flatten the out-of-transit portion of each light curve. Furthermore, the differential light curves were derived using an optimal photometric aperture and a set of comparison stars chosen by the observer.

Numerous ground-based facilities conducted photometric follow-up of the TOI-732 system. Their respective data acquisition and reduction strategies are described in the following sections, while their detrended light curves are plotted in Figure 2. Differences in the instrumental setups and nightly observing conditions produce varying levels of photometric precision among the light curves. Each detrended light curve, available through TFOP, is fit with a Mandel & Agol (2002) transit model that we calculate using the batman software package (Kreidberg 2015). The shallow transit depths of both planet candidates produce low-S/N transit light curves that may only marginally improve the measurement precision on most model parameters compared to the values measured from the TESS light curve, with the exception being the planets' orbital periods when all light curves are fit simultaneously. As such, we fix the orbital periods and impact parameters in the individual light-curve fits to the values obtained from the SPOC data validation module (P_b = 0.76842 days, P_c = 12.25422 days, b_b = 0.69, b_c = 0.35). We also derive the scaled semimajor axes using the stellar parameters given in Table 1 (a_b/R_s = 6.96, a_c/R_s = 44.09). Each planet's orbit is also fixed to circular, and the quadratic limb-darkening parameters in the corresponding passband are interpolated from the Claret & Bloemen (2011) tables using the EXOFAST software (Eastman et al. 2013) given LTT 3780's ${T}_{\mathrm{eff}}$, $\mathrm{log}g$, and [Fe/H]. We fit the following parameters via nonlinear least-squares optimization using scipy.curve_fit: the baseline flux f₀, the time of mid-transit T₀, and the planet-to-star radius ratio r_p/R_s. Measuring T₀ relative to the expected transit time is used to refine the planet's orbital ephemeris, while r_p/R_s measurements in each passband are required to investigate transit depth chromaticity, as a chromatically varying transit depth could be indicative of a blended EB.

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3.3.1. LCOGT Photometry

3.3.2. OSN Photometry

3.3.3. TRAPPIST-North Photometry

3.3.4. MEarth-North Photometry

Very nearby stars that are not detected in Gaia DR2 or any of the seeing-limited image sequences and that fall within the same 21'' TESS pixel as the target star will result in photometric contamination that is unaccounted for in the TESS light curve. This effect reduces the depth of the observed transits and can produce a false-positive transit signal from another astrophysical source, such as a blended EB (Ciardi et al. 2015). We used two independent sets of high-resolution follow-up imaging sequences to search for any such close-in sources, as described in the following sections.

3.4.1. SOAR Speckle Imaging

We obtained SOAR speckle imaging (Tokovinin 2018) of LTT 3780 on UT 2019 December 12 in the I band, a visible bandpass similar to that of TESS. Details of the observations from the SOAR TESS survey are provided in Ziegler et al. (2020). No bright nearby stars are detected within 3'' of LTT 3780 within the 5σ detection sensitivity of the observations. The resulting 5σ contrast curve is plotted in Figure 3 along with the speckle autocorrelation function.

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3.4.2. NIRI AO Imaging

3.5.1. HARPS RVs

3.5.2. HARPS-N RVs

We begin by analyzing the TESS PDCSAP light curve wherein the planet candidates TOI-732.01 and 02 were initially detected. The majority of apparent signals from nonrandom noise sources in the light curve have already been removed by the SPOC processing. However, low-frequency and small-amplitude signals that do not resemble planetary transits are seen to persist in the PDCSAP light curve shown in Figure 4. The nature of these signals as residual systematics or photometric stellar variability is unclear, so we proceed with modeling the aforementioned noise signals as an untrained semiparametric Gaussian process (GP) regression model simultaneously with the two transiting planet candidates using the exoplanet software package (Foreman-Mackey et al. 2019). This software computes analytical transit models using the STARRY package (Luger et al. 2019) and uses the celerite package (Foreman-Mackey et al. 2017) to evaluate the marginalized likelihood under a GP model. In this analysis, the covariance kernel takes the form of a stochastically driven, damped simple harmonic oscillator (SHO) whose Fourier transform is known as the power spectral density (PSD) and is given by

$$\begin{eqnarray}&&S(\omega )=\sqrt{\displaystyle \frac{2}{\pi }}\displaystyle \frac{{S}_{0}\,{\omega }_{0}^{4}}{{\left({\omega }^{2}-{{\omega }_{0}}^{2}\right)}^{2}+{{\omega }_{0}}^{2}\,{\omega }^{2}/{Q}^{2}}.\end{eqnarray} \tag{ 1 }$$

The PSD of the SHO is parameterized by the frequency of the undamped oscillator ω₀, S₀, which is proportional to the power at the frequency ω₀, and the quality factor Q, which is fixed to $\sqrt{0.5}$. We selected this covariance kernel and parameterization because working in Fourier space is much more computationally efficient for large data sets, such as our TESS light curve (N = 15,210), and because the underlying cause of the photometric variations being modeled remains unknown. In practice, we also fit for the baseline flux f₀ and an additive scalar jitter s_TESS. We fit the GP hyperparameters using the parameter combinations $\{\mathrm{ln}{\omega }_{0},\mathrm{ln}{S}_{0}{\omega }_{0}^{4},{f}_{0},\mathrm{log}{s}_{\mathrm{TESS}}^{2}\}$ with uninformative priors.

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The transit model within exoplanet fits the stellar mass M_s, stellar radius R_s, and quadratic limb-darkening parameters {u₁, u₂} along with the following planetary parameters: logarithmic orbital periods $\mathrm{ln}P$, times of mid-transit T₀, logarithmic planet radii $\mathrm{ln}{r}_{p}$, impact parameters b, and eccentricity and argument of periastron of LTT 3780c only {e_c, ω_c}. We assume a circular orbit for the inner planet LTT 3780b because its ultrashort period of 0.77 days implies a very short circularization timescale of ≪1 Myr (Goldreich & Soter 1966). Jointly fitting for the physical stellar and planetary parameters enables us to derive the transit observables a/R_s, r_p/R_s and inclination i. The joint GP plus two-planet transit model therefore includes 18 model parameters: $\{{f}_{0}$, $\mathrm{ln}{\omega }_{0}$, $\mathrm{ln}{S}_{0}{\omega }_{0}^{4}$, $\mathrm{ln}{s}_{\mathrm{TESS}}^{2}$, ${M}_{s},{R}_{s},$ ${u}_{1}$, ${u}_{2}$, $\mathrm{ln}{P}_{b}$, ${T}_{0,b}$, $\mathrm{ln}{r}_{p,b}$, ${b}_{b}$, $\mathrm{ln}{P}_{c}$, ${T}_{0,c}$, $\mathrm{ln}{r}_{p,c}$, b_c, e_c, ω_c}. Table 3 summarizes the TESS transit model parameter priors used in our fiducial analysis.

Table 3.  TESS Light-curve and RV Model Parameter Priors

Parameter	Fiducial Model Priors	EXOFASTv2 Model Priors
Stellar Parameters
${T}_{\mathrm{eff}}$ [K]	${ \mathcal N }(3331,157)$	${ \mathcal N }(3351,150)$
Ms [M⊙]	${ \mathcal N }(0.401,0.012)$	${ \mathcal N }(0.401,0.012)$
Rs [R⊙]	${ \mathcal N }(0.374,0.011)$	${ \mathcal N }(0.374,0.011)$
Light-curve Hyperparameters
f0	${ \mathcal N }(0,10)$	${ \mathcal U }(-\inf ,\inf )$
$\mathrm{ln}\,{\omega }_{0}$ [day−1]	${ \mathcal N }(0,10)$	⋯
$\mathrm{ln}\,{S}_{0}{\omega }_{0}^{4}$	${ \mathcal N }(\mathrm{ln}\,\mathrm{var}({{\boldsymbol{f}}}_{\mathrm{TESS}}),10)$	⋯
$\mathrm{ln}\,{s}_{\mathrm{TESS}}^{2}$	${ \mathcal N }(\mathrm{ln}\,\mathrm{var}({{\boldsymbol{f}}}_{\mathrm{TESS}}),10)$	⋯
u1	${ \mathcal U }(0,1)$	${ \mathcal U }(0.225,0.425)$
u2	${ \mathcal U }(0,1)$	${ \mathcal U }(0.232,0.432)$
Dilution	⋯	${ \mathcal N }(0,0.1\ \delta )$ a
RV Parameters
$\mathrm{ln}\,\lambda$ [days]	${ \mathcal U }(\mathrm{ln}\,1,\mathrm{ln}1000)$	⋯
$\mathrm{ln}\,{\rm{\Gamma }}$	${ \mathcal U }(-3,3)$	⋯
$\mathrm{ln}\,{P}_{\mathrm{GP}}$ [days]	${ \mathcal N }(\mathrm{ln}104,\mathrm{ln}30)$ b	⋯
$\mathrm{ln}\,{a}_{\mathrm{HARPS}}$ [m s−1]	${ \mathcal U }(-5,5)$	⋯
$\mathrm{ln}\,{a}_{\mathrm{HARPS} \mbox{-} {\rm{N}}}$ [m s−1]	${ \mathcal U }(-5,5)$	⋯
$\mathrm{ln}\,{s}_{\mathrm{HARPS}}$ [m s−1]	${ \mathcal U }(-5,5)$	${ \mathcal U }(-\inf ,\inf )$
$\mathrm{ln}\,{s}_{\mathrm{HARPS} \mbox{-} {\rm{N}}}$ [m s−1]	${ \mathcal U }(-5,5)$	${ \mathcal U }(-\inf ,\inf )$
${\gamma }_{\mathrm{HARPS}}$ [m s−1]	${ \mathcal U }(-185,205)$	${ \mathcal U }(-\inf ,\inf )$
γHARPS-N [m s−1]	${ \mathcal U }(-185,205)$	${ \mathcal U }(-\inf ,\inf )$
LTT 3780b Parameters
$\mathrm{ln}\,{P}_{b}$ [days]	${ \mathcal N }(\mathrm{ln}0.768,0.5)$	⋯
Pb [days]	⋯	${ \mathcal U }(-\inf ,\inf )$
T0,b [BJD–2,457,000]	${N}(\mathrm{1,543.911},0.5)$	${ \mathcal U }(\mathrm{1,543.7},1544.2)$
$\mathrm{ln}\,{r}_{p,b}$ [R⊕]	${ \mathcal N }(0.5\cdot \mathrm{ln}({Z}_{b})+\mathrm{ln}\,{R}_{s},1)$ c	⋯
rp,b/Rs	⋯	${ \mathcal U }(-\inf ,\inf )$
bb	${ \mathcal U }(0,1+{r}_{p,b}/{R}_{s})$	⋯
$\mathrm{ln}\,{K}_{b}$ [m s−1]	${ \mathcal U }(-5,5)$	⋯
Kb [m s−1]	⋯	${ \mathcal U }(-\inf ,\inf )$
LTT 3780c Parameters
$\mathrm{ln}\,{P}_{c}$ [days]	${ \mathcal N }(\mathrm{ln}12.254,0.5)$	⋯
Pc [days]	⋯	${ \mathcal U }(-\inf ,\inf )$
T0,c [BJD–2,457,000]	${ \mathcal N }(\mathrm{1,546.848},0.5)$	${ \mathcal U }(\mathrm{1,542.8},1550.9)$
$\mathrm{ln}\,{r}_{p,c}$ [R⊕]	${ \mathcal N }(0.5\cdot \mathrm{ln}({Z}_{c})+\mathrm{ln}\,{R}_{s},1)$ d
rp,c/Rs	⋯	${ \mathcal U }(-\inf ,\inf )$
bc	${ \mathcal U }(0,1+{r}_{p,c}/{R}_{s})$
$\mathrm{ln}\,{K}_{c}$ [m s−1]	${ \mathcal U }(-5,5)$	⋯
Kc [m s−1]	⋯	${ \mathcal U }(-\inf ,\inf )$
ec	${ \mathcal B }(0.867,3.03)$ e
ωc [rad]	${ \mathcal U }(-\pi ,\pi )$

Notes. Gaussian distributions are denoted by ${ \mathcal N }$ and parameterized by mean and standard deviation values. Uniform distributions are denoted by ${ \mathcal U }$ and bounded by the specified lower and upper limits. Beta distributions are denoted by ${ \mathcal B }$ and parameterized by the shape parameters α and β.

^aδ is the SPOC-derived dilution factor applied to the TESS light curve. ^bP_GP is constrained by the estimate of the stellar rotation period from $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$, whose uncertainty is artificially inflated. ^cThe transit depth of TOI-732.01 reported by the SPOC: Z_b = 1253 ppm. ^dThe transit depth of TOI-732.02 reported by the SPOC: Z_b = 3417 ppm. ^eKipping (2013).

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Our full model is fit to the TESS PDCSAP light curve using the PyMC3 Markov Chain Monte Carlo (MCMC) package (Salvatier et al. 2016) implemented within exoplanet. We ran four simultaneous chains with 4000 tuning steps and 3000 draws in the final sample. PyMC3 produces the 18-dimensional joint posterior probability density function (PDF) of the model parameters. Median point estimates from the marginalized posterior PDFs of the GP hyperparameters are used to construct the GP predictive distribution whose mean function is shown in Figure 4 and used to detrend the TESS light curve for visualization purposes. Similarly, the median point estimates of the transit model parameters are used to compute the "best-fit" transit models shown in Figure 4. Table 4 reports the median values of all model parameters from their marginalized posterior PDFs along with their uncertainties from the 16th and 84th percentiles.

Table 4.  Point Estimates of the LTT 3780 Planetary System Model Parameters

Parameter	Fiducial Model Valuesa	EXOFASTv2 Model Valuesb
TESS Light-curve Parameters
Baseline flux, f0	1.000072 ± 0.000070	1.000043 ± 0.000038
$\mathrm{ln}\,{\omega }_{0}$	1.64 ± 1.15	⋯
$\mathrm{ln}\,{S}_{0}{\omega }_{0}^{4}$	${3.62}_{-0.39}^{+0.40}$	⋯
$\mathrm{ln}\,{s}_{\mathrm{TESS}}^{2}$	1.21 ± 0.01	⋯
TESS limb-darkening coefficient, u1	${0.28}_{-0.20}^{+0.33}$	${0.30}_{-0.05}^{+0.07}$
TESS limb-darkening coefficient, u2	${0.16}_{-0.28}^{+0.37}$	${0.32}_{-0.06}^{+0.07}$
Dilution	⋯	${0.023}_{-0.048}^{+0.047}$
RV Parameters
$\mathrm{ln}\,\lambda \,{\mathrm{day}}^{-1}$	${4.5}_{-0.4}^{+1.0}$	⋯
$\mathrm{ln}\,{\rm{\Gamma }}$	$-{0.1}_{-1.2}^{+1.3}$	⋯
$\mathrm{ln}\,{P}_{\mathrm{GP}}\,{\mathrm{day}}^{-1}$	${4.64}_{-0.16}^{+0.14}$	⋯
$\mathrm{ln}\,{a}_{\mathrm{HARPS}}/{\mathrm{ms}}^{-1}$	${0.52}_{-0.62}^{+0.69}$	⋯
$\mathrm{ln}\,{a}_{\mathrm{HARPS} \mbox{-} {\rm{N}}}/{\mathrm{ms}}^{-1}$	${1.25}_{-0.74}^{+0.70}$	⋯
Jitter, sHARPS [m s−1]	${0.11}_{-0.09}^{+0.48}$	${1.41}_{-0.80}^{+0.70}$
Jitter, sHARPS-N [m s−1]	${1.24}_{-0.46}^{+0.36}$	${3.54}_{-0.75}^{+0.99}$
Systemic velocity, γHARPS [m s−1]	${195.5}_{-1.5}^{+1.4}$	${195.4}_{-0.5}^{+0.5}$
Systemic velocity, γHARPS-N [m s−1]	${196.8}_{-3.6}^{+4.6}$	${194.3}_{-1.0}^{+1.0}$
LTT 3780b (TOI-732.01) Parameters
Log orbital period, $\mathrm{ln}\,{P}_{b}$	−0.26338 ± 0.00007	⋯
Orbital period, Pb [days]	${0.768448}_{-0.000053}^{+0.000055}$	${0.7683881}_{-0.0000083}^{+0.0000084}$
Time of mid-transit, T0,b [BJD–2,457,000]	1,543.9115 ± 0.0011	${\mathrm{1,543.91199}}_{-0.00051}^{+0.00059}$
Transit duration, Db [hr]	${0.805}_{-0.072}^{+0.049}$	${0.786}_{-0.020}^{+0.024}$
Transit depth, Zb [ppt]	${1.087}_{-0.103}^{+0.098}$	${1.076}_{-0.089}^{+0.093}$
Scaled semimajor axis, ab/Rs	${7.03}_{-0.21}^{+0.23}$	${7.05}_{-0.22}^{+0.24}$
Planet-to-star radius ratio, rp,b/Rs	${0.0330}_{-0.0016}^{+0.0014}$	${0.0328}_{-0.0014}^{+0.0014}$
Impact parameter, bb	${0.35}_{-0.23}^{+0.20}$	${0.43}_{-0.12}^{+0.08}$
Inclination, ib [deg]	${87.1}_{-1.7}^{+1.8}$	${86.5}_{-0.7}^{+1.0}$
Eccentricity, eb	0 (fixed)	0 (fixed)
Planet radius, rp,b [R⊕]	${1.332}_{-0.075}^{+0.072}$	${1.321}_{-0.073}^{+0.074}$
Log RV semi-amplitude, $\mathrm{ln}\,{K}_{b}$	${1.23}_{-0.17}^{+0.14}$	${1.26}_{-0.17}^{+0.14}$
RV semi-amplitude, Kb [m s−1]	${3.41}_{-0.63}^{+0.63}$	${3.54}_{-0.55}^{+0.54}$
Planet mass, mp,b [M⊕]	${2.62}_{-0.46}^{+0.48}$	${2.77}_{-0.43}^{+0.43}$
Bulk density, ρb [g cm−3]	${6.1}_{-1.5}^{+1.8}$	${6.5}_{-1.4}^{+1.7}$
Surface gravity, gb [m s−2]	${14.4}_{-3.3}^{+3.7}$	${15.5}_{-3.4}^{+3.6}$
Escape velocity, vesc,b [km s−1]	${15.7}_{-1.5}^{+1.5}$	${16.2}_{-1.4}^{+1.3}$
Semimajor axis, ab [au]	${0.01211}_{-0.00012}^{+0.00012}$	${0.01212}_{-0.00012}^{+0.00012}$
Insolation, Fb [F⊕]	${106}_{-19}^{+22}$	${106}_{-19}^{+23}$
Equilibrium temperature, Teq,b [K]
Bond albedo = 0.0	892 ± 44	892 ± 44
Bond albedo = 0.3	816 ± 40	816 ± 40
LTT 3780c (TOI-732.02) Parameters
Log orbital period, $\mathrm{ln}\,{P}_{c}$	2.50582 ± 0.00023	⋯
Orbital period, Pc [days]	${12.2519}_{-0.0030}^{+0.0028}$	${12.252048}_{-0.000059}^{+0.000060}$
Time of mid-transit, T0,c [BJD–2,457,000]	1,546.8484 ± 0.0014	${\mathrm{1,546.8481}}_{-0.0012}^{+0.0011}$
Transit duration, Dc [hr]	${1.392}_{-0.049}^{+0.050}$	${1.404}_{-0.046}^{+0.048}$
Transit depth, Zc [ppt]	${3.24}_{-0.37}^{+0.41}$	${3.13}_{-0.28}^{+0.28}$
Scaled semimajor axis, ac/Rs	${44.6}_{-1.3}^{+1.5}$	${44.7}_{-1.4}^{+1.5}$
Planet-to-star radius ratio, rp,c/Rs	${0.0570}_{-0.0033}^{+0.0035}$	${0.0560}_{-0.0025}^{+0.0024}$
Impact parameter, bc	${0.65}_{-0.36}^{+0.15}$	${0.71}_{-0.15}^{+0.08}$
Inclination, ic [deg]	${89.18}_{-0.22}^{+0.47}$	${88.95}_{-0.09}^{+0.10}$
${e}_{c}\cos {\omega }_{c}$	⋯	$-{0.05}_{-0.08}^{+0.07}$
${e}_{c}\sin {\omega }_{c}$	⋯	${0.15}_{-0.13}^{+0.15}$
$\sqrt{{e}_{c}}\cos {\omega }_{c}$	${0.13}_{-0.15}^{+0.12}$	⋯
$\sqrt{{e}_{c}}\sin {\omega }_{c}$	${0.07}_{-0.19}^{+0.17}$	⋯
Eccentricity, ec	${0.06}_{-0.14}^{+0.15}$	${0.18}_{-0.11}^{+0.14}$
Argument of periastron, ωc [deg]	${124}_{-147}^{+87}$	${111}_{-27}^{+39}$
Planet radius, rp,c [R⊕]	${2.30}_{-0.15}^{+0.16}$	${2.25}_{-0.13}^{+0.13}$
Log RV semi-amplitude, $\mathrm{ln}\,{K}_{c}$	${1.49}_{-0.17}^{+0.17}$	${1.60}_{-0.15}^{+0.13}$
RV semi-amplitude, Kc [m s−1]	${4.44}_{-0.68}^{+0.82}$	${4.94}_{-0.67}^{+0.68}$
Planet mass, mp,c [M⊕]	${8.6}_{-1.3}^{+1.6}$	${9.5}_{-1.3}^{+1.3}$
Bulk density, ρc [g cm−3]	${3.9}_{-0.9}^{+1.0}$	${4.6}_{-0.9}^{+1.1}$
Surface gravity, gc [m s−2]	${16.0}_{-3.3}^{+3.7}$	${18.3}_{-3.1}^{+3.5}$
Escape velocity, ${v}_{\mathrm{esc},c}$ [km s−1]	${21.7}_{-2.0}^{+2.1}$	${23.0}_{-1.7}^{+1.7}$
Semimajor axis, ac [au]	${0.07673}_{-0.00077}^{+0.00075}$	${0.07678}_{-0.00077}^{+0.00076}$
Insolation, Fc [F⊕]	${2.63}_{-0.48}^{+0.56}$	${2.63}_{-0.48}^{+0.56}$
Equilibrium temperature, ${T}_{\mathrm{eq},c}$ [K]
Bond albedo = 0.0	353 ± 18	354 ± 18
Bond albedo = 0.3	323 ± 16	324 ± 16

Notes.

^aOur fiducial model features sequential modeling of the TESS light curve, with an SHO GP detrending component plus two transiting planets, followed by the RV analysis conditioned on the results of the transit analysis. The fiducial RV model includes a quasi-periodic activity model plus two Keplerian orbital solutions. The LTT 3780b Keplerian component is fixed to a circular orbit. ^bOur alternative analysis is a global model of the TESS light curve, ground-based light curves, and RVs using the EXOFASTv2 software. The input light curves have already been detrended, and the residual RV noise is treated as an additive scalar jitter. This global model produces self-consistent results between the transit and RV data set and improves the precision on each planet's orbital ephemeris by including the ground-based transit light curves.

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In our fiducial analysis, we elected to fit the RVs independently of the transit data but exploiting the strong priors on the orbital periods and mid-transit times established by our TESS light-curve analysis (Section 4.1). We note that the information content within the TESS light curve and the RV measurements with regards to their shared model parameters (i.e., {P_b, T_0,b, P_c, T_0,c, e_c, ω_c}) is dominated by one data set or the other. In other words, the strongest constraints on each planet's orbital period and mid-transit time are derived from the TESS and ground-based transit light curves. Conversely, most of the information regarding the eccentricity and argument of periastron of LTT 3780c is derived from the RVs, since the planet's secondary eclipse is unresolved in the TESS light curve and the eccentricity's effect on the transit duration is degenerate with a/R_s, r_p/R_s, and b. Note that this is only an approximation, as global transit plus RV modeling can help to mitigate the eccentricity degeneracy (Eastman et al. 2019). We will also consider a global model in Section 4.3.

Although LTT 3780 is known to be relatively inactive, we do not expect its surface to be completely static and homogeneous. As such, we expect some temporally correlated residual RV signals from magnetic activity that we model with a quasi-periodic GP regression model for each spectrograph. The quasi-periodic covariance kernel is

$$\begin{eqnarray}&&{k}_{{ij}}={a}^{2}\exp \,\left[-\displaystyle \frac{{\left({t}_{i}-{t}_{j}\right)}^{2}}{2{\lambda }^{2}}-{{\rm{\Gamma }}}^{2}{\sin }^{2}\,\left(\displaystyle \frac{\pi | {t}_{i}-{t}_{j}| }{{P}_{\mathrm{GP}}}\right)\right]\end{eqnarray} \tag{ 2 }$$

and features four hyperparameters: the covariance amplitude a, exponential timescale λ, coherence Γ, and periodic timescale P_GP. We also fit an additive scalar jitter s_RV for each spectrograph to absorb any excess white noise. Due to the unique systematic noise properties of each spectrograph, we fit a unique covariance amplitude and scalar jitter to the data from each of the HARPS and HARPS-N spectrographs. Throughout, the covariance parameters {λ, Γ, P_GP}, which only depend on signals originating from the star, are kept fixed between the two spectrographs.

Our full RV model consists of a GP activity model for each spectrograph plus independent Keplerian orbital solutions for each planet with RV semi-amplitudes K_b and K_c. We also fit for each spectrograph's systemic velocity γ to account for any RV offset between the two instruments. Our full RV model therefore features 17 model parameters: $\{\mathrm{ln}{a}_{\mathrm{HARPS}}$, $\mathrm{ln}{a}_{\mathrm{HARPS} \mbox{-} {\rm{N}}}$, $\mathrm{ln}\lambda$, $\mathrm{ln}{\rm{\Gamma }}$, $\mathrm{ln}{P}_{\mathrm{GP}},$ $\mathrm{ln}{s}_{\mathrm{HARPS}}$, $\mathrm{ln}{s}_{\mathrm{HARPS} \mbox{-} {\rm{N}}}$, γ_HARPS, ${\gamma }_{\mathrm{HARPS} \mbox{-} {\rm{N}}},$ P_b, T_0,b, $\mathrm{ln}{K}_{b},$ P_c, T_0,c, $\mathrm{ln}{K}_{c}$, h_c, k_c}, where ${h}_{c}=\sqrt{{e}_{c}}\cos {\omega }_{c}$ and ${k}_{c}=\sqrt{{e}_{c}}\sin {\omega }_{c}$. Note that the GP hyperparameters, scalar jitter parameters, and planetary semi-amplitudes are fit in logarithmic units. Table 3 includes each of the RV model parameter priors.

Figure 5 shows the raw RVs and individual model components, including the RV activity, along with LTT 3780b and c. The Bayesian generalized Lomb–Scargle periodogram (BGLS; Mortier et al. 2015) of each RV component is also included in Figure 5. The BGLS of the raw RVs exhibits a small number of significant peaks (e.g., 3.1 days) that are not strictly at either planet's orbital period. We will see that the subtraction of the individual Keplerian orbits effectively removes these periodicities such that they can be attributed to harmonics of the planetary orbital periods. The median RV model parameters from their marginalized posterior PDFs are used to produce the models shown in Figure 5 and are reported in Table 4 along with their 16th and 84th percentiles. The RV semi-amplitudes of LTT 3780b and c are found to be ${3.41}_{-0.63}^{+0.63}$ and ${4.44}_{-0.68}^{+0.82}$ m s⁻¹ and thus are clearly detected at 5.4σ and 5.9σ, respectively. The resulting Keplerian RV signals are clearly discernible in their phase-folded RV time series. The rms values of the RV residuals are found to be 1.55 and 1.74 m s⁻¹ for HARPS and HARPS-N, respectively.

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The M dwarfs are known to commonly host two to three planets per star out to 200 days (e.g., Dressing & Charbonneau 2015; Ballard & Johnson 2016; Hardegree-Ullman et al. 2019; Cloutier & Menou 2020) such that the probability that a third planet exists around LTT 3780 is nonnegligible. However, the BGLS of the RV residuals in Figure 5 does not exhibit any strong periodic signals that are statistically significant. This indicates that a hypothetical third planet is unlikely to have been detected. To confirm this robustly, we considered a three-planet RV model with fixed Keplerian parameters for LTT 3780b and c plus a third Keplerian component "d" on a circular orbit. We separately tested two three-planet models with differing priors on the orbital period P_d: ${ \mathcal U }(1.3,2.1)$ and ${ \mathcal U }(50,150)$ days. The chosen period limits approximately span the two highest peaks in the BGLS of the RV residuals. We then ran two separate MCMCs to sample the posteriors of the hypothetical planet's period, time of inferior conjunction (analogous to the mid-transit time), and semi-amplitude. We find that neither model settles on a preferred period or phase, and each marginalized P_d posterior simply recovers its uninformative prior. The lack of a well-defined maximum a posteriori P_d and T_0,d prevents us from searching the TESS light curve for any missed transit signals from the hypothetical planet "d" and from placing a meaningful upper limit on the planet's mass. We note that the only threshold crossing events identified by the TPS in the TESS light curve were those corresponding to the confirmed planets LTT 3780b and c. Additionally, the recovered semi-amplitudes K_d in both MCMCs favored 0 m s⁻¹ with an upper limit of K_d ≤ 2.4 m s⁻¹ at 95% confidence. Taken together, these findings emphasize that the fiducial two-planet model for the current data set is likely complete, as no third planet is detected in our data.

To evaluate the robustness of the results derived in our fiducial analysis (Sections 4.1 and 4.2), we conducted an independent analysis using the EXOFASTv2 exoplanet transit plus RV fitting package (Eastman et al. 2019). The methods of the EXOFASTv2 fitting routine are detailed in Eastman et al. (2019), although we provide a brief summary here.

To constrain the stellar-dependent parameters during the transit fit, we feed EXOFASTv2 the M_s and R_s parameter priors as in our fiducial model. The routine also takes as input the pre-detrended light curves from TESS and ground-based facilities and performs a differential MCMC to evaluate the two-planet transit model whose parameter priors are included in Table 3.

There are a few notable differences between our fiducial analysis (Sections 4.1 and 4.2) and the EXOFASTv2 approach. The exoplanet model simultaneously fits the hyperparameters of the GP detrending model plus the transiting planet parameters to achieve self-consistent detrending and transit models wherein the uncertainties in the recovered planet parameters are marginalized over uncertainties in the detrending model. Conversely, EXOFASTv2 uses pre-detrended light curves, so the aforementioned marginalization of the planet parameter uncertainties over the GP hyperparameters does not occur. Furthermore, the RV model in our fiducial analysis includes the treatment of residual RV signals as a quasi-periodic GP, whereas EXOFASTv2 assumes the RV residuals to be well represented by a Gaussian noise term characterized by an additive jitter factor.

Our EXOFASTv2 modeling has the important advantage of evaluating a global model that includes the TESS light curve, ground-based transit light curves, and RV measurements. The resulting planet parameters, including the orbital periods, mid-transit times, eccentricities, and argument of periastron, will therefore be self-consistent between all input data sets. In particular, each planet's ephemeris will be more precisely constrained by the inclusion of the ground-based transit light curves, and the eccentricity of LTT 3780c will be jointly constrained by its transit duration, Keplerian RV model, and stellar density. EXOFASTv2 also fits a free dilution parameter to model any discrepancies between the dilution applied to the PDCSAP light curve and the true dilution.

The results from our fiducial model in Table 4 are accompanied by the results from our alternative analysis using EXOFASTv2. We find consistency between the two models at <1σ for nearly all model parameters. This speaks to the robustness of the planetary model parameters inferred from our data. The only exceptions are the 2σ and 2.8σ discrepant RV jitter parameters s_HARPS and s_HARPS-N. However, this is not alarming, as the RV residuals, following the removal of the two Keplerian solutions, are modeled with a GP in our fiducial model, whereas the EXOFASTv2 model treats the residuals with a scalar jitter. Crucially, these approaches yield consistent RV semi-amplitudes for LTT 3780b and c whose agreement between the two models is 0.2σ and 0.7σ, respectively.

From our analysis of the TESS transit light curve, we measure the planetary radii of LTT 3780b and c to be ${r}_{p,b}={1.332}_{-0.075}^{+0.072}$ and ${r}_{p,c}={2.30}_{-0.15}^{+0.16}$ R_⊕. By combining the TESS analysis with the mid-transit times measured from transit follow-up observations, we measure orbital periods for LTT 3780b and c to be ${P}_{b}={0.7683881}_{-0.0000083}^{+0.0000084}$ and ${P}_{c}={12.252048}_{-0.000059}^{+0.000060}$ days. This places LTT 3780b at 0.012 au, where it receives 106 times Earth's insolation. Assuming uniform heat redistribution and a Bond albedo of zero, LTT 3780b has an equilibrium temperature of T_eq,b = 892 K. Similarly, the orbital period of LTT 3780c places it at 0.077 au, where it receives 2.6 times Earth's insolation with a zero-albedo equilibrium temperature of 353 K.

From our RV analysis, we measure planet masses of ${m}_{p,b}={2.62}_{-0.46}^{+0.48}$ and ${m}_{p,c}={8.6}_{-1.3}^{+1.6}$ M_⊕, which represent 5.6σ and 5.9σ mass detections, respectively. By combining the planetary mass and radius measurements, we derive bulk densities of ${\rho }_{p,b}={6.1}_{-1.5}^{+1.8}$ and ${\rho }_{p,c}={3.9}_{-0.9}^{+1.0}$ g cm⁻³. Figure 6 details the mass–radius diagram of exoplanets around M dwarfs with masses measured at the level of ≥3σ, including the LTT 3780 planets. The LTT 3780 planet masses and radii are compared to theoretical models of fully differentiated planetary interiors consisting of combinations of water, silicate rock, and iron (Zeng & Sasselov 2013). In Figure 6, we see that LTT 3780b is consistent with an Earth-like bulk composition of 33% iron plus 67% magnesium silicate by mass. This composition is shared by the majority of planets in the ≲1.5 R_⊕ size regime. We also consider models of Earth-like solid cores that include 1% H₂ envelopes by mass over a range of equilibrium temperatures from 300 to 1000 K (Zeng et al. 2019). The mass and radius of LTT 3780c appear consistent with a water-dominated bulk composition but also with a predominantly Earth-like body that hosts an extended low mean molecular weight atmosphere. Distinguishing between these two degenerate structure models will require the extent of LTT 3780c's atmosphere to be investigated through transmission spectroscopy. Due to the dependence of the atmospheric scale height on the planet's surface gravity, the accurate interpretation of forthcoming transmission spectroscopy observations will be facilitated by the planetary mass measurements presented in this study. The feasibility of targeting LTT 3780c with transmission spectroscopy is discussed in Section 5.4.

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The LTT 3780 two-planet system adds to the growing number of confirmed multiplanet systems around nearby M dwarfs with at least one transiting planet (e.g., GJ 1132, Berta-Thompson et al. 2015; K2-3, Crossfield et al. 2015; K2-18, Montet et al. 2015; LHS 1140, Dittmann et al. 2017; TRAPPIST-1, Gillon et al. 2017; Bonfils et al. 2018; Damasso et al. 2018; L98-59, Kostov et al. 2019; Ment et al. 2019; LP 791-18, Crossfield et al. 2019; TOI-270, Günther et al. 2019; Cloutier et al. 2019a; Cloutier et al. 2019b; TOI-700, Gilbert et al. 2020; Rodriguez et al. 2020). With their sub-Neptune-sized radii and measured masses presented herein, both LTT 3780b and c contribute directly to the completion of the TESS level-one science requirement to obtain masses for 50 planets smaller than 4 R_⊕.

The occurrence rate distribution of close-in planet radii around Sun-like stars features a bimodality with a dearth of planets at 1.7–2.0 R_⊕ known as the radius valley (Fulton et al. 2017; Mayo et al. 2018). This feature likely results from the existence of a transition between predominantly rocky planets and larger planets that host significant H/He envelopes as a function of planet radius and orbital separation. The slope of the radius valley in P–r_p space marks the critical radius separating rocky and nonrocky planets as a function of orbital period. The empirical slope of the radius valley around Sun-like stars is consistent with models of thermally driven atmospheric mass loss such as photoevaporation and core-powered mass loss (van Eylen et al. 2018; Martinez et al. 2019; Wu 2019). However, for mid-K to mid-M dwarfs, the radius valley slope flattens and becomes increasingly favored by models of an alternative formation pathway for terrestrial planets in a gas-poor environment (CM20).

Figure 7 depicts the LTT 3780 planets in P–r_p space along with the subset of M dwarf planets from Figure 6 with RV-derived masses. The planets in Figure 7 are classified as having a bulk composition that is either rocky, gaseous, or intermediate based on their mass and radius. Rocky planets are defined as planets that are consistent with having a bulk density greater than that of 100% MgSiO₃ given their size. Similarly, unambiguously gaseous planets are defined as planets that are consistent with having a bulk density less than that of 100% H₂O given their size. The remaining planets are flagged as having bulk compositions that are intermediate between rocky and gaseous. Planets LTT 3780b and c have rocky and intermediate dispositions, respectively (Figure 6).

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In Figure 7, LTT 3780b and c are shown to span the empirically derived location of the radius valley around low-mass stars under the gas-poor formation and photoevaporation models (CM20). The slope of the radius valley around low-mass stars is considerably flatter than around Sun-like stars, with the former slope being consistent with gas-poor formation while the latter is more consistent with a thermally driven atmospheric mass-loss process. To compare the compositions of planets around low-mass stars to the rocky/nonrocky transition locations in Figure 7, we scale the transition measured around Sun-like stars down to the low stellar mass regime under the photoevaporation model (${r}_{p}\propto {({M}_{s}/{M}_{\odot })}^{1/4};$ Wu 2019).⁴⁷ The slope measured around low-mass stars is plotted verbatim in Figure 7. Both models predict that LTT 3780b should have a rocky bulk composition in which any residual gaseous envelope only contributes marginally to the planet's mass and radius. Indeed, these predictions are consistent with our finding that LTT 3780b has an Earth-like composition. Similarly, both models predict that LTT 3780c should be nonrocky in that it should have retained a substantial gaseous envelope and therefore be inconsistent with having a bulk rocky composition. Although we cannot definitively identify the bulk composition of LTT 3780c with our data, due to internal structure model degeneracies, we confirm that LTT 3780c is consistent with both model predictions. In other words, the models correctly identify LTT 3780c as being inconsistent with an Earth-like composition and requiring a significant amount of volatile material or H/He gas to explain its mass and radius.

5.2.1. Planetary Mass Limits from Photoevaporation Models

Stars such as LTT 3780 with multitransiting planets that span the radius valley provide valuable test cases of radius valley emergence models. The virtue of these systems is that limits on the planetary masses can be derived by scaling the properties of one planet to the other (Owen & Campos Estrada 2020). For example, assuming that the initial H/He envelope of the rocky planet below the valley has been completely stripped by some physical process, the theoretical minimum mass of the nonrocky planet above the valley can be calculated by scaling its properties to those of the rocky planet. An equivalent principle can be used to derive the maximum mass of the rocky planet. The power of this comparative scaling of planets within the same planetary system is that certain unobservable quantities that directly affect final planet masses are scaled out. An example of this is the host star's XUV luminosity history in the photoevaporation scenario (Owen & Campos Estrada 2020).

A full derivation is presented in Appendix A but here we simply state the condition for the consistency of the gaseous (i.e., nonrocky) and rocky planet parameters with the photoevaporation model. This requires the gaseous planet's mass-loss timescale to exceed the maximum mass-loss timescale of the rocky planet (Owen & Campos Estrada 2020). This condition leads to

$$\begin{eqnarray}&&1\leqslant \displaystyle \frac{{m}_{\mathrm{core},\mathrm{gas}}^{0.64}}{{m}_{\mathrm{core},\mathrm{rock}}}{\left(\displaystyle \frac{{a}_{\mathrm{gas}}}{{a}_{\mathrm{rock}}}\right)}^{2/3}{r}_{\mathrm{core},\mathrm{rock}}^{4/3},\end{eqnarray} \tag{ 3 }$$

where each planet's core mass and radius are given in units of the Earth. In the LTT 3780 system, we define LTT 3780b to be the rocky planet below the valley whose H/He envelope has been photoevaporated away, leaving behind a solid core whose mass and radius are equal to the planet's total mass and radius: m_core,rock = m_p,b = 2.62 ± 0.47 M_⊕ and r_core,rock = r_p,b = 1.332 ± 0.074 R_⊕. The gaseous planet above the valley is then LTT 3780c, whose mass is assumed to be dominated by an Earth-like core such that m_core,gas = m_p,c = 8.6 ± 1.5 M_⊕ and whose core radius is approximated by the mass–radius relation for Earth-like bodies (${r}_{p}\propto {m}_{p}^{1/3.7};$ Zeng et al. 2016). Lastly, the semimajor axes a_rock and a_gas are a_b = 0.01211 ± 0.00012 and a_c = 0.07673 ± 0.00076 au, respectively.

Using Equation (3) and sampling the planetary parameters Θ = {m_p,b, a_b, r_p,b, a_c} from their marginalized posterior PDFs, we find that the mass of LTT 3780c must be ≥0.49 ± 0.15 M_⊕ in order to be consistent with the photoevaporation model. In the same way, but by replacing m_p,b with m_p,c in the set Θ, we calculate that the mass of LTT 3780b must be ≤19.6 ± 2.8 M_⊕ to be consistent with photoevaporation. Clearly, the measured masses m_p,c = 8.6 ± 1.5 and m_p,b = 2.62 ± 0.47 M_⊕ are both consistent with predictions from the photoevaporation model, implying that photoevaporation is a feasible process for sculpting the observed architecture of the LTT 3780 system.

A few notable caveats exist with the planetary mass limits imposed by the photoevaporation model in Equation (3) (Owen & Campos Estrada 2020). These are discussed in Appendix A.

5.2.2. Planetary Mass Limits from Core-powered Mass-loss Models

Similarly to the photoevaporation model, we can compare the mass-loss timescales of the LTT 3780 planets under the core-powered mass-loss scenario (Ginzburg et al. 2018; Gupta & Schlichting 2019, 2020) to constrain their permissible planet masses under that model. In the core-powered mass-loss scenario, the lower atmosphere is in thermal contact with the planetary core, which conducts energy from its formation into the atmosphere. This heat flux drives convective heat transport radially outward to the radiative–convective boundary (RCB) of the atmosphere, above which the atmosphere is isothermal at ${T}_{\mathrm{eq}}$ and atmospheric cooling is radiative. The physical limit of the atmospheric mass-loss rate is given by the thermal velocity of the gas at the Bondi radius, the radial distance at which the escape velocity equals the thermal sound speed ${c}_{s}=\sqrt{{k}_{{\rm{B}}}{T}_{\mathrm{eq}}/\mu }$, where k_B is the Boltzmann constant and μ is the atmospheric mean molecular weight, which we fix to 2 amu for H₂.

The derivation of the mass-loss timescale in the core-powered mass-loss model is presented in Appendix B. As in the photoevaporation scenario, we require the mass-loss timescale for the gaseous planet to exceed that of the rocky planet, which leads to the following condition for consistency of the planetary parameters with the core-powered mass-loss model:

$$\begin{eqnarray}&&\begin{array}{l}1\leqslant \left(\displaystyle \frac{{m}_{\mathrm{core},\mathrm{gas}}}{{m}_{\mathrm{core},\mathrm{rock}}}\right){\left(\displaystyle \frac{{T}_{\mathrm{eq},\mathrm{gas}}}{{T}_{\mathrm{eq},\mathrm{rock}}}\right)}^{-3/2}\\ \ \times \ \exp \left[c^{\prime} \left(\displaystyle \frac{{m}_{\mathrm{core},\mathrm{gas}}}{{T}_{\mathrm{eq},\mathrm{gas}}\ {r}_{p,\mathrm{gas}}}-\displaystyle \frac{{m}_{\mathrm{core},\mathrm{rock}}}{{T}_{\mathrm{eq},\mathrm{rock}}\ {r}_{p,\mathrm{rock}}}\right)\right],\end{array}\end{eqnarray} \tag{ 4 }$$

where the constant $c^{\prime} =G\mu /{k}_{{\rm{B}}}\approx {10}^{4}$ R_⊕ K M_⊕⁻¹, T_eq,gas = T_eq,c = 323 ± 16 K, T_eq,rock = T_eq,b = 816 ± 40 K, r_p,gas = r_p,c = 2.30 ± 0.16 R_⊕, and r_p,rock = r_p,b = 1.332 ± 0.074 R_⊕. The inequality in Equation (4) has no analytic solution, so we solve for the limiting masses of m_core,gas and m_core,rock by again sampling the planetary parameters {m_core,rock, T_eq,rock, r_p,rock, m_core,gas, T_eq,gas, r_p,gas} from their marginalized posterior PDFs and numerically solving for the limiting core masses. Recall that both planets are assumed to have small envelope mass fractions such that m_core ≈ m_p.

Under the core-powered mass-loss mechanism, we find that the mass of LTT 3780c must be ≥2.1 ± 0.5 M_⊕ to be consistent with the model. Similarly, by solving for m_core,rock, we calculate that the mass of LTT 3780b must be ≤12.6 ± 2.9 M_⊕. As with the photoevaporation mass limits from Section 5.2.1, the measured masses m_p,c = 8.6 ± 1.5 and m_p,b = 2.62 ± 0.47 M_⊕ are both consistent with predictions from the core-powered mass-loss model.

The masses of LTT 3780b and c recovered in this study from HARPS and HARPS-N RV measurements are both consistent with radius valley emergence model predictions from photoevaporation and core-powered mass loss, two physical processes that thermally drive atmospheric escape on close-in planets. Thus, the recovered masses of LTT 3780b and c are unable to provide strong evidence for the inapplicability of either mechanism. However, the photoevaporation and core-powered mass-loss models do make distinct predictions for the maximum mass of the rocky planet and the minimum mass of the nonrocky in systems like LTT 3780 that feature such planet pairs. Therefore, other systems with multitransiting planets that span the radius valley may exist for which either photoevaporation or core-powered mass loss may be ruled out by the planets' masses. This prospect is especially viable for increasingly compact systems wherein the ratios a_gas/a_rock and T_eq,gas/T_eq,rock approach unity.

5.2.3. Planetary Mass Limits from Gas-poor Terrestrial Planet Formation Models

We used the TTV2Fast2Furious python package (Hadden 2019) to predict the amplitudes of the transit timing variations (TTVs) of the planets LTT 3780b and c. We ran 10³ realizations with the planetary masses being sampled from their marginalized posterior PDFs from our RV analysis (Section 4.2). The stellar mass, planet orbital periods, and times of mid-transit are drawn from their respective priors used in our RV analysis. Recall that the free eccentricity of LTT 3780b is assumed to be zero because of its short circularization timescale. Furthermore, due to their large period ratio (P_b = 0.768388 days, P_c = 12.252048 days, P_c/P_b = 15.945130), imposing a nonzero free eccentricity on either planet will have a negligible effect on their TTV amplitudes, so we fix the input free eccentricities to zero. The forced eccentricities induced by the planets' mutual interactions are calculated within TTV2Fast2Furious. Arguments of periastron are drawn from ${ \mathcal U }(0,2\pi )$.

In each realization, with its unique set of parameters, we compute each planet's maximum deviation from a linear ephemeris over a 2 yr baseline beginning with the commencement of the TESS sector 9 observations. Over the 10³ realizations, we find maximum TTV amplitudes of 0.02 and 1 s for LTT 3780b and c, respectively. The small amplitude of the expected TTV signals makes the LTT 3780 system a poor candidate for intensive transit follow-up to derive the TTV masses of the two known planets. However, ongoing transit observations of LTT 3780c may reveal TTVs induced by an insofar-unseen outer planet. For this purpose, we note that LTT 3780 is scheduled to be observed in sector 35 of the TESS extended mission between UT 2021 February 9 and March 7.

The stellar and planetary parameters of the LTT 3780 system make the planets LTT 3780b and c accessible targets for atmospheric characterization via emission and transmission spectroscopy, respectively. Assuming uniform heat redistribution and a Bond albedo of zero, the equilibrium temperature of LTT 3780c is T_eq,c = 353 K. The expected depth of its transmission features up to two atmospheric scale heights (Stevenson 2016; Fu et al. 2017) in a cloud-free low mean molecular weight atmosphere (μ = 2) is 79 ppm. Alternatively, it is expected that some mini-Neptune atmospheres are metal-enriched (Fortney et al. 2013), which will partially suppress transmission feature depths to 32 ppm in a 100× solar metallicity atmosphere (μ ≈ 5). Simulated transit observations with PandExo (Batalha et al. 2017) confirm that molecular features in a clear, low mean molecular weight atmosphere will be detectable at ≥5σ confidence from a single transit observation with JWST/NIRISS slitless spectroscopy⁴⁸ (Kreidberg et al. 2015). Four transits would be required to reach a similar precision for a 100× solar metallicity atmosphere. We also note the caveat that if high-altitude clouds are present on LTT 3780c, as seen for many other planets in its size regime (Crossfield & Kreidberg 2017), additional observing time will be required.

For LTT 3780c, we can also consider the transmission spectroscopy metric (TSM; Kempton et al. 2018), which is proportional to the expected S/N of transmission features in a cloud-free atmosphere. Based on the TSM, LTT 3780c is among the best warm mini-Neptunes (P ∈ [10, 40] days, r_p ∈ [2, 3] R_⊕) for atmospheric characterization via transmission spectroscopy observations. To date, the best such planets are the TESS-discovered planets TOI-700c (Gilbert et al. 2020; Rodriguez et al. 2020), TOI-270d (Günther et al. 2019), and LTT 3780c, whose TSM values are all within 17% of each other and at minimum 17% greater than that of the next best potential target: HD 15337c (Dumusque et al. 2019). The TSM values of favorable warm mini-Neptunes are reported in Table 5 and compared in Figure 8.

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Table 5.  TSM Values for Warm Mini-Neptunes^a

Planet	P	rp	mp	Z	Teqb	J	${T}_{\mathrm{eff}}$	Rs	Ms	TSM	TSM	References
Name	(days)	(R⊕)	(M⊕)	(ppt)	(K)	(mag)	(K)	(R⊙)	(M⊙)		Normalized
TOI-270d	11.38	2.13	5.48c	2.6	372	9.099	3386	0.38	0.40	86.8	1.00	1
TOI-700 c	16.05	2.63	7.64c	3.3	356	9.469	3480	0.42	0.42	77.5	0.89	2, 3
LTT 3780c	12.25	2.30	8.59	3.3	353	9.007	3331	0.37	0.40	71.5	0.82	4
HD 15337c	17.17	2.52	8.79	0.6	648	7.553	5125	0.87	0.90	60.6	0.70	5
GJ 143b	35.61	2.61	22.70	1.2	427	6.081	4640	0.70	0.73	53.0	0.61	6
K2-266d	14.70	2.93	8.90	1.5	538	9.611	4285	0.70	0.69	47.1	0.54	7
K2-18b	32.94	2.71	8.63	2.8	290	9.763	3505	0.47	0.50	42.8	0.49	8
Kepler-96b	15.24	2.67	8.46	0.6	798	9.260	5690	1.02	1.00	30.6	0.35	9
K2-266e	19.48	2.73	14.30	1.3	490	9.611	4285	0.70	0.69	21.3	0.24	7
Kepler-102e	16.15	2.22	8.93	0.7	604	9.984	4909	0.76	0.81	16.3	0.19	9
HD 119130b	16.98	2.63	24.50	0.5	801	8.730	5725	1.09	1.00	11.3	0.13	10
K2-38 c	10.56	2.42	9.90	0.3	928	9.911	5757	1.38	2.24	9.2	0.11	11

Notes.

^aHere we define warm mini-Neptunes as having P∈[10, 40] days and r_p ∈ [2, 3] R_⊕. ^bHere T_eq is calculated assuming zero albedo and full heat redistribution. ^cPlanet masses are estimated using the mass–radius relation implemented in the forecaster code (Chen & Kipping 2017).

References. (1) Günther et al. (2019), (2) Gilbert et al. (2020), (3) Rodriguez et al. (2020), (4) this work, (5) Dumusque et al. (2019), (6) Dragomir et al. (2019), (7) Rodriguez et al. (2018), (8) Cloutier et al. (2019b), (9) Marcy et al. (2014), (10) Luque et al. (2019), (11) Sinukoff et al. (2016).

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The ultrashort-period planet LTT 3780b has a zero-albedo equilibrium temperature of T_eq,b = 892 K. The hot dayside of LTT 3780b makes it a very attractive target for atmospheric characterization via emission spectroscopy observations. In particular, eclipse observations can help to discern whether the planet has retained a substantial atmosphere or if its emitting temperature is consistent with that of pure rock. The distinction between a 1 bar atmosphere and a bare rocky surface on LTT 3780b will be accessible with a single JWST/MIRI eclipse observation (Koll et al. 2019).

Similarly to the TSM, the expected S/N of thermal emission signatures at 7.5 μm is proportional to the emission spectroscopy metric (ESM; Kempton et al. 2018). Computing the ESM for hot planets with likely terrestrial compositions (r_p < 1.5 R_⊕) that are favorable targets for emission spectroscopy measurements reveals that LTT 3780b is among the best such planets (Table 6, Figure 8). The ESM for LTT 3780b is the third-highest among these planets and closely matches that of GJ 1252b (Shporer et al. 2020). Both of these targets have ESM values that are nearly half that of LHS 3844b (Vanderspek et al. 2019), a rocky planet whose thermal phase curve has been characterized by the Spitzer Space Telescope and found to be consistent with a dark basaltic surface that lacks any substantial atmosphere (Kreidberg et al. 2019).

Table 6.  ESM Values for Select Close-in Earth-sized Planets^a

Planet	P	rp	Z	Teqb	Tdayc	Ks	${T}_{\mathrm{eff}}$	Rs	Ms	ESM	ESM	References
Name	(days)	(R⊕)	(ppt)	(K)	(K)	(mag)	(K)	(R⊙)	(M⊙)		Normalized
LHS 3844b	0.46	1.30	4.0	805	886	9.145	3036	0.19	0.15	29.0	1.00	1
GJ 1252b	0.52	1.19	0.8	1089	1198	7.915	3458	0.39	0.38	16.4	0.57	2
LTT 3780b	0.77	1.33	1.1	892	982	8.204	3331	0.37	0.40	13.4	0.46	3
L168-9b	1.40	1.39	0.5	981	1079	7.082	3800	0.60	0.62	9.9	0.34	4
GJ 1132b	1.63	1.13	2.4	585	643	8.322	3270	0.21	0.18	9.5	0.33	5
L98-59c	3.69	1.35	1.6	515	566	7.101	3412	0.31	0.31	6.9	0.24	6
LTT 1445Ab	5.36	1.38	2.0	435	478	6.500	3335	0.28	0.26	6.3	0.22	7
LP 791-18b	0.95	1.12	3.6	594	653	10.644	2949	0.17	0.14	5.9	0.20	8
L98-59b	2.25	0.80	0.6	607	668	7.101	3412	0.31	0.31	4.1	0.14	6
TRAPPIST-1b	1.51	1.09	6.9	405	446	10.300	2559	0.12	0.08	4.0	0.14	9
LHS 1140c	3.78	1.28	3.1	434	477	8.821	3216	0.21	0.18	3.4	0.12	10

Notes.

^aHere we define Earth-sized planets as those with r_p < 1.5 R_⊕. ^bHere T_eq is calulated assuming zero albedo and full heat redistribution. ^cFor the purpose of calculating ESM values, we assume that T_day = 1.1T_eq for all planets.

References. (1) Vanderspek et al. (2019), (2) Shporer et al. (2020), (3) this work, (4) Astudillo-Defru et al. (2020), (5) Berta-Thompson et al. (2015), (6) Kostov et al. (2019), (7) Winters et al. (2019), (8) Crossfield et al. (2019), (9) Gillon et al. (2017), (10) Ment et al. (2019).

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The favorable ESM and TSM values of LTT 3780b and c, respectively, make them both accessible targets for atmospheric characterization. Together they present a unique opportunity to conduct direct comparative studies of exoplanet atmospheres among planets within the same planetary system, which is critical for informing our understanding of the formation and evolution of close-in planets at a range of sizes and equilibrium temperatures.

Following the announcement of the planet candidates TOI-732.01 and 02 in 2019 May, multiple precision RV instrument teams began working toward the mass characterization of these potential planets. This study has presented the subset of those efforts from HARPS and HARPS-N, but we acknowledge that the CARMENES team has also submitted a paper presenting their own RV time series and analysis (Nowak et al. 2020). Although the submissions of these complementary studies were coordinated between the two groups, their respective data, analyses, and write-ups were intentionally conducted independently.