Add the hidden-companions paper, the double-lined paper's wider HARPS sample, NIRPS companions and eclipse masses, and nine searches
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| \title{Double-Lined from the Archive:\\ Independent Dynamical Masses of M Dwarfs in Eclipsing Binaries\\ from Public HARPS and ESPRESSO Spectra} | |
| \author{Charles C. Norton} | |
| \date{} | |
| \begin{document} | |
| \maketitle | |
| \begin{abstract} | |
| An eclipsing binary of a solar-type star and an M dwarf gives model-independent masses for both | |
| stars once the M dwarf's own lines are measured, at optical flux ratios near $10^{-3}$. This paper | |
| measures them from public ESO archive spectra with an analysis built independently of the published | |
| ones. The spectra are placed in the primary's frame at their measured velocities and divided by | |
| their series' median, cross-correlated with an empirical M1\,V spectrum, cleaned of the star's and | |
| the sky's patterns and of five principal components, and summed along the companion's track over | |
| its semi-amplitude \Ktwo\ and its systemic offset. Each summit is set against $10^4$ draws of two | |
| nulls, the orbit's phase moved and whole nights shuffled, with a Gumbel law for the tail, and the | |
| 35 binaries of the survey are accounted for together. TOI-1338/BEBOP-1 (260 ESPRESSO spectra), EBLM | |
| J0310$-$31 (114 HARPS spectra) and HD\,4875 (35 HARPS spectra) give false-alarm probabilities of | |
| $10^{-11}$, $8\times10^{-12}$ and $1\times10^{-8}$ and \Ktwo\ $= 77.27 \pm 0.10 \pm 0.15$, $88.12 | |
| \pm 0.18 \pm 0.34$ and $75.29 \pm 0.30 \pm 0.87$\,\kms\ (statistical, then systematic from the | |
| series' parts by date and its seasons), against published values of $77.16 \pm 0.48$, $88.27 \pm | |
| 0.46$ and $75.13 \pm 0.16$\,\kms; HD\,4875's companion had been measured only in the | |
| near-infrared. The companion's lines sit $+1.05$ to $+1.25$\,\kms\ from the primary's systemic | |
| velocity. Among the 35 southern EBLM binaries with archived HARPS series, one further companion | |
| stands with a false-alarm probability of $1.5\times10^{-7}$ across the whole survey: EBLM | |
| J2101$-$45, whose 75 spectra give \Ktwo\ $= 65.41 \pm 0.32 \pm 0.32$\,\kms\ and minimum masses of | |
| $1.415$ and $0.552$\,\Msun, stable under changes of template, band and principal components and | |
| recovered when a companion is injected into the same spectra. J0649$-$27, J0825$-$67 and J1141$-$37 | |
| reach 3.3 to $3.9\sigma$ alone but 2.1 to $3.0\sigma$ across the survey, and remain candidates. | |
| Among 41 further EBLM binaries whose orbits are fitted to their own HARPS velocities, the | |
| companions of J2122$-$32, J0425$-$46, J2054$-$32 and J0535$-$08 stand at 5.3 to $5.7\sigma$ across | |
| the 41 and four more are candidates. NIRPS spectra recover J1540$-$09's published companion at | |
| \Ktwo\ $= 59.98 \pm 0.50 \pm 0.78$\,\kms\ and give a candidate in J1613$-$76, whose HARPS series | |
| shows none, and the TESS eclipses turn nine of these binaries' minimum masses into masses and | |
| radii, J0535$-$08's $0.970 \pm 0.011$ and $0.521 \pm 0.003$\,\Msun. | |
| \end{abstract} | |
| \section{Introduction}\label{sec:intro} | |
| Masses and radii of M dwarfs set the properties inferred for the planets that orbit them, and | |
| the empirical relations behind those inferences rest on stars whose masses are measured | |
| without models. Detached double-lined eclipsing binaries provide such masses, but few are | |
| known below the fully convective boundary near 0.35\,\Msun~\cite{Torres2010,Southworth2015}, | |
| since an M dwarf paired with a solar-type star contributes about a thousandth of the optical | |
| light and its lines are lost in the primary's. The EBLM project has characterised hundreds of | |
| these single-lined eclipsing binaries~\cite{Triaud2013,Triaud2017,Maxted2023}, measuring the | |
| primary's spectroscopic orbit and the eclipses and taking the primary's mass or radius from | |
| stellar models. The BEBOP survey for circumbinary planets has added years of precise velocities | |
| of the same stars with HARPS and SOPHIE~\cite{Martin2019,Standing2022,Triaud2022}. | |
| The secondary's lines can nonetheless be recovered. High-resolution cross-correlation, developed | |
| to detect molecules in exoplanet atmospheres~\cite{Snellen2010,Brogi2012}, sums a weak spectrum | |
| along a known orbit after the stationary spectrum has been removed. Sebastian et al.\ applied it | |
| to the circumbinary planet host TOI-1338/BEBOP-1~\cite{Kostov2020,Standing2023} and detected its | |
| M dwarf at $11\sigma$ in 103 ESPRESSO spectra, measuring \Ktwo\ $= 77.16 \pm 0.48$\,\kms\ and | |
| masses of $1.098 \pm 0.017$ and $0.307 \pm 0.003$\,\Msun~\cite{Sebastian2024b}, with the | |
| Saltire model of the cross-correlation map~\cite{Sebastian2024a}. The method has since been | |
| applied to Kepler-16 with SOPHIE~\cite{Sebastian2025b}, to BEBOP-3~\cite{Baycroft2025} and to | |
| five more EBLM binaries, among them EBLM J0310$-$31~\cite{Stevenson2026}. In the near-infrared, | |
| where the M dwarf is relatively brighter, the secondary's semi-amplitude has been measured with | |
| SPIRou~\cite{Maxted2022}, CRIRES+~\cite{Sebastian2025a} and NIRPS~\cite{Adshead2026,Hahlin2026}. | |
| These measurements share a small number of analysis designs, and their accuracy has been tested | |
| by splitting each dataset in time and by simulation~\cite{Sebastian2024b,Stevenson2026}. An | |
| analysis built independently, with a different template, a different removal of the primary's | |
| spectrum and a different measure of significance, tests that accuracy from outside. The public | |
| archives also hold more spectra of these stars than the published analyses used: the ESO archive | |
| holds 114 HARPS spectra of EBLM J0310$-$31 above a signal-to-noise ratio of 10, against the 71 of | |
| ref.~\cite{Stevenson2026}, and 264 ESPRESSO spectra of TOI-1338, against 103. | |
| This paper describes such an analysis, the binaries approach of the seti-tools package, which | |
| shares its spectra machinery with the package's planet-frame and free-orbit companion searches and | |
| their orbitlight release~\cite{orbitlight}, and applies it to public products of the ESO Science | |
| Archive alone. Section~\ref{sec:data} selects the targets and spectra and | |
| Section~\ref{sec:method} describes the method, its statistic and its calibration. | |
| Section~\ref{sec:validation} compares its results for three systems with published dynamical | |
| masses, which test its accuracy, its uncertainties and its false-alarm probabilities. | |
| Section~\ref{sec:survey} applies it to the southern EBLM binaries with HARPS series and separates | |
| the one new measurement from the candidates. Section~\ref{sec:wider} extends it to EBLM binaries | |
| whose orbits come from their own spectra, to NIRPS spectra and to the TESS eclipses, and | |
| Section~\ref{sec:discussion} discusses what the comparison shows. | |
| \section{Targets and spectra}\label{sec:data} | |
| The targets are the single-lined spectroscopic binaries with archived high-resolution spectra. | |
| The Gaia DR3 non-single-star catalogue~\cite{GaiaNSS2023} gives 203\,121 single-lined orbits | |
| (solution types SB1, SB1C and AstroSpectroSB1) for stars brighter than $G = 13$, and the SB9 | |
| catalogue~\cite{Pourbaix2004} 3\,426 single-lined orbits. Each was matched within | |
| 10\,arcsec to the ESO archive's series of 20 or more spectra of one target from HARPS, ESPRESSO, | |
| NIRPS, UVES or FEROS, giving 140 series of 126 stars. A series is searched when its orbit has a | |
| period below 100\,d covered three or more times, the series holds 30 or more spectra, and the | |
| mass function allows a companion of 0.07\,\Msun\ or more around a primary of 1\,\Msun. That | |
| leaves 68 series. Among them are the HARPS series of 37 EBLM binaries, named here by their EBLM | |
| designations, which this paper searches, and TOI-1338's ESPRESSO series; the others belong to O and | |
| B stars, interacting binaries with compact companions, and solar-type binaries observed mostly with | |
| UVES or FEROS. | |
| The spectra are the archive's Phase 3 one-dimensional products, with barycentric wavelengths and | |
| the pipeline's flux and error. For each series, every product with a signal-to-noise ratio above | |
| 10 is taken, up to 300 spread evenly over the series' span. | |
| \section{Method}\label{sec:method} | |
| \subsection{The primary's velocities and orbit} | |
| Each spectrum is placed on a grid uniform in velocity at 1\,\kms\ from 3800 to 7880\,\AA, | |
| divided by its running median over 501 samples, with the telluric bands removed. Its velocity is | |
| the peak of its cross-correlation over $\pm 200$\,\kms\ with the line depths of a G0\,V template, | |
| one minus the median of EBLM J0310$-$31's 114 HARPS spectra placed at their velocities on its | |
| Gaia orbit, refined by a parabola through the three highest samples. The primary's orbit is fitted | |
| to these velocities in two steps. A grid over period (within 0.4 per cent of the catalogue's, in | |
| steps that move the last epoch by a fiftieth of an orbit), phase (in hundredths), eccentricity (to | |
| 0.6) and argument of periastron (in $20^\circ$ steps) is searched with \Kone\ and the systemic | |
| velocity $\gamma$ solved linearly at each point, and the best point starts a least-squares fit with | |
| a soft loss in period, time of periastron, $\sqrt{e}\cos\omega$, $\sqrt{e}\sin\omega$, \Kone\ and | |
| $\gamma$. The grid makes the fit independent of errors in the catalogue orbit, which for | |
| TOI-1338 put the period 0.021\,d long and $\omega$ $58^\circ$ away. | |
| \subsection{Residuals in the primary's frame} | |
| Each spectrum is then moved into the primary's frame at its own measured velocity, placed on the | |
| same grid, divided by its running median and by the series' median spectrum, and a cubic in | |
| wavelength is removed. A residual shift of a spectrum against the median of 0.5\,\kms\ or more is | |
| taken out before the division. The telluric bands and the sodium sky lines at their telescope | |
| wavelengths are masked. Each wavelength's scatter across the series, floored at a tenth of its | |
| median, weights the cross-correlation. | |
| \subsection{Cross-correlation with an M-dwarf template} | |
| The template is an empirical M1\,V spectrum: the line depths, one minus the median, of the archived | |
| HARPS spectra of GJ\,367~\cite{Lam2021} in its own frame, negative values set to zero. Each | |
| residual is cross-correlated with it by inverse variance over $\pm 300$\,\kms\ at 1\,\kms, the | |
| result normalised so that a residual of unit noise gives unit scatter. Each cross-correlation row | |
| then loses three patterns that are not the companion's: the star's own residual pattern, the | |
| median row fitted to each row within 40\,\kms\ of the star and subtracted; the sky's pattern, the | |
| rows' median in the telluric frame fitted back to each row; and five principal components of the | |
| rows, which carry the shapes the star's pattern takes from night to night. Each row is weighted by | |
| the inverse square of its robust scatter more than 50\,\kms\ from the star. | |
| \subsection{The sum along the companion's track} | |
| The companion's barycentric velocity at time $t$ is | |
| \begin{equation} | |
| v_2(t) = \gamma + \Delta v_\mathrm{sys} - K_2\left[\cos(\nu + \omega) + e\cos\omega\right], | |
| \end{equation} | |
| with $\nu$, $\omega$ and $e$ from the primary's fitted orbit and \dvsys\ the offset of the | |
| companion's lines from the primary's systemic velocity. With $C_i(v)$ the cleaned | |
| cross-correlation of spectrum $i$ at velocity $v$ in the frame in which it was placed, | |
| $v_{\star,i}$ the velocity at which it was placed and $w_i$ its weight, the trail's value is | |
| \begin{equation} | |
| S(K_2, \Delta v_\mathrm{sys}) = \frac{\sum_i w_i\, m_i\, C_i\!\left(v_2(t_i) - v_{\star,i}\right)}{\sum_i w_i\, m_i}, | |
| \end{equation} | |
| read by linear interpolation, where $m_i$ is one when the reading lies more than 30\,\kms\ from the | |
| primary's lines and more than 10\,\kms\ from the row's telluric velocity and zero otherwise. $S$ is | |
| defined where the readings number at least a fifth of the rows and at least ten. The map of $S$ | |
| over \Ktwo\ from \Kone\ to 160\,\kms\ and over \dvsys\ from $-30$ to $+30$\,\kms, both at | |
| 0.5\,\kms, gives the peak within 5\,\kms\ of zero offset, refined on grids of 0.1 and then | |
| 0.05\,\kms. | |
| \subsection{The statistic and its false-alarm probability}\label{sec:stat} | |
| The search statistic is the summit $\hat S = \max_{K_2} S$ over a grid from \Kone\ to 250\,\kms\ | |
| in steps of 0.25\,\kms, either at a fixed offset or, with the offset free, over the offsets within | |
| 5\,\kms\ of zero in steps of 0.5\,\kms\ as well. A null draw computes the same summit with the | |
| companion's track replaced, by the orbit's phase moved by a uniform 0.1 to 0.9 of a period or by | |
| whole nights moved to other nights' dates, each night keeping its spectra's offsets from its start, | |
| so that what is tied to the orbital phase and what is tied to the hour or the night both stay in a | |
| null. Each binary is set against $10^4$ draws of each null, $\hat S^{(j)}$, and three numbers | |
| describe its summit. The first is its height in units of the nulls' scatter, | |
| \begin{equation} | |
| z = \frac{\hat S - \langle \hat S^{(j)} \rangle}{\mathrm{sd}(\hat S^{(j)})}, | |
| \end{equation} | |
| a standardised statistic rather than a probability. The second is the empirical false-alarm | |
| probability, $p_\mathrm{emp} = (1 + n_\ge)/(10^4 + 1)$, with $n_\ge$ the draws at or above $\hat | |
| S$, which cannot fall below $10^{-4}$. The third extends the nulls with a tail model. Each null | |
| summit is the largest of many correlated values of the trail, so the summits follow an | |
| extreme-value law, and the Gumbel law~\cite{Gumbel1958} with the nulls' mean and scatter gives | |
| \begin{equation} | |
| p_\mathrm{G} = 1 - \exp\!\left\{-\exp\!\left[-\frac{\hat S - \mu}{\beta}\right]\right\},\qquad | |
| \beta = \frac{\sqrt{6}}{\pi}\,\mathrm{sd}(\hat S^{(j)}),\quad \mu = \langle \hat S^{(j)} \rangle - 0.5772\,\beta . | |
| \end{equation} | |
| The null of the two with the larger $p_\mathrm{G}$ is adopted, and $p_\mathrm{G}$ is quoted with | |
| the one-sided Gaussian significance of the same probability. A generalised extreme-value law fitted | |
| to the same draws~\cite{Coles2001} measures the tail's shape $\xi$: zero for Gumbel's, negative for | |
| a lighter tail, for which $p_\mathrm{G}$ overstates the probability, and positive for a heavier one. | |
| The 35 searched binaries are accounted for together. In each draw of a null, the highest of the 35 | |
| binaries' summits, each in units of its own nulls' scatter, gives one draw of the survey's highest | |
| null, and each binary's $z$ is set against these draws with a Gumbel law fitted in the same way. | |
| Beside it, each binary's own $p_\mathrm{G}$ is compounded over the 35~\cite{Sidak1967}, | |
| $1 - (1 - p_\mathrm{G})^{35}$. The larger of the two, under the null that gives the larger, is the | |
| false-alarm probability across the survey. A companion is taken as detected when its own | |
| $p_\mathrm{G}$ reaches $5\sigma$ ($2.9\times10^{-7}$), and the probability across the survey is | |
| quoted beside it. | |
| \subsection{Uncertainty and masses}\label{sec:uncertainty} | |
| The statistical uncertainty of \Ktwo\ is half the 16th to 84th percentile range of 300 draws of | |
| whole nights with replacement, each cleaned again and its peak found on grids of 0.25 and then | |
| 0.05\,\kms. Noise that persists over a season is not in it. As in the published | |
| analyses~\cite{Sebastian2024b,Stevenson2026}, each series is divided by date into four parts of | |
| about equal numbers of spectra (two parts below 60 spectra), whole nights kept together, each part | |
| cleaned and measured alone, and the parts' standard error with Bessel's correction is a systematic | |
| uncertainty. The series is also resampled by observing season, runs of nights without a gap | |
| longer than 90\,d, in 300 draws of whole seasons with replacement when it spans four seasons or | |
| more. The larger of the two is the systematic uncertainty, added in quadrature to the statistical | |
| one for the masses, which follow from | |
| \begin{equation} | |
| M_{1,2}\sin^3 i = 1.0361\times10^{-7}\,(1-e^2)^{3/2}\,(K_1+K_2)^2\,K_{2,1}\,P , | |
| \end{equation} | |
| with the velocities in \kms, the period in days and the masses in \Msun, divided by $\sin^3 i$ at | |
| the published inclination from the eclipses. | |
| \subsection{Robustness tests}\label{sec:robust} | |
| Each companion is measured again under five changes of the analysis: three and eight principal | |
| components in place of five; the M-dwarf template kept only blueward or only redward of 5500\,\AA; | |
| and the template made instead from the 110 archived HARPS spectra of L\,98-59, an M3 dwarf with | |
| the same instrument~\cite{Cloutier2019}, in the same way as GJ\,367's. The band and template | |
| changes rebuild the rows from the spectra. A synthetic companion is also laid into the actual | |
| spectra before any processing: the template's lines at a V-band flux ratio of $10^{-3}$, and for | |
| the four companions of Section~\ref{sec:survey} also $3\times10^{-4}$, scaled in wavelength as the | |
| ratio of 3400 and 6000\,K black bodies, moving at the measured \Ktwo\ and an offset of $+1.0$\,\kms\ | |
| on the primary's orbit but a quarter of a period later than the real companion, so that the two | |
| tracks cross only twice an orbit. The injected spectra go through the whole analysis and are read | |
| along the moved track, the null with the phase moved kept a tenth of a period from the real | |
| companion's track. The ratio of the real companion's peak to the injected one's at $10^{-3}$ gives | |
| the flux ratio at which the real companion's lines match the template's. | |
| \section{Three systems with published dynamical masses}\label{sec:validation} | |
| Three of the binaries have published measurements of the secondary's semi-amplitude from other | |
| data or other methods: TOI-1338 (EBLM J0608$-$59), measured from ESPRESSO spectra with | |
| Saltire~\cite{Sebastian2024b}; EBLM J0310$-$31, measured from HARPS spectra with | |
| Saltire~\cite{Stevenson2026}; and HD\,4875 (EBLM J0050$-$18), measured from NIRPS spectra by | |
| cross-correlation with a synthetic M-dwarf spectrum~\cite{Hahlin2026}. The archive holds 264 | |
| ESPRESSO spectra of TOI-1338 above the signal-to-noise cut, of which 260 load, 114 HARPS spectra of | |
| J0310$-$31 and 35 of HD\,4875. They test the method's accuracy, its uncertainties and the | |
| calibration of its false-alarm probabilities on companions whose semi-amplitudes are known. | |
| \subsection{Orbits} | |
| The orbits fitted to the measured velocities (Table~\ref{tab:orbits}) leave a scatter of 14, 15 | |
| and 29\,m\,s$^{-1}$, which is the precision of a cross-correlation velocity against a template | |
| built from one star's spectra, and they agree with the published orbits to within | |
| $0.012$\,\kms\ in \Kone\ and $0.0003$ in eccentricity. The systemic velocities differ from the | |
| published ones by 0.01 to 0.14\,\kms, the offset between the template's zero point and those of the | |
| published analyses, which enters the companion's track only through \dvsys. | |
| \begin{table}[t] | |
| \centering | |
| \caption{The primary's orbits fitted to the velocities measured here, against the published orbits. | |
| $\omega$ is the primary's argument of periastron; rms is the scatter of the velocities about the | |
| fit.}\label{tab:orbits} | |
| \begin{tabular}{llrrrrr} | |
| \toprule | |
| System & Source & $P$ (d) & \Kone\ (\kms) & $e$ & $\omega$ ($^\circ$) & rms (m\,s$^{-1}$)\\ | |
| \midrule | |
| TOI-1338 & this work & 14.608560 & 21.616 & 0.1556 & 117.86 & 14\\ | |
| & ref.~\cite{Sebastian2024b} & 14.6085579 & 21.604 & 0.1555 & 117.77 & \\ | |
| J0310$-$31 & this work & 12.642791 & 27.868 & 0.3089 & 185.83 & 15\\ | |
| & ref.~\cite{Stevenson2026} & 12.6427926 & 27.875 & 0.3087 & 185.85 & \\ | |
| HD\,4875 & this work & 13.635615 & 29.736 & 0.1814 & 251.71 & 29\\ | |
| & ref.~\cite{Hahlin2026} & 13.6355611 & 29.739 & 0.1811 & 252.03 & \\ | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \subsection{The companions} | |
| The companion stands out in each system's map (Fig.~\ref{fig:maps}) at the published \Ktwo\ and | |
| just redward of the primary's systemic velocity, and the trail over \Ktwo\ at that offset rises | |
| alone above the nulls (Fig.~\ref{fig:curves}). Table~\ref{tab:k2} sets the results against the | |
| published ones. The semi-amplitudes differ from the published values by $+0.11$, $-0.15$ and | |
| $+0.16$\,\kms, which is 0.2, 0.3 and 0.2 times the two uncertainties combined, and at the published | |
| inclinations the masses agree within 0.6 per cent. HD\,4875's published value comes from | |
| near-infrared spectra alone. | |
| The summits stand 19.3, 19.4 and 13.8 times the nulls' scatter above their mean, where the highest | |
| of the adopted null's $10^4$ draws reaches 3.1, 5.2 and 3.0 times. No draw comes within reach of | |
| the observed summits, so the empirical probabilities are at their floor of $10^{-4}$, and | |
| the Gumbel law gives $10^{-11}$, $8\times10^{-12}$ and $1\times10^{-8}$, or 6.7, 6.7 and | |
| $5.6\sigma$ (Table~\ref{tab:calib}). With the offset free within 5\,\kms\ of zero they are | |
| $2\times10^{-12}$, $3\times10^{-10}$ and $2\times10^{-7}$. Across the survey's 35 binaries, | |
| J0310$-$31 stays at $3\times10^{-10}$ ($6.2\sigma$) and HD\,4875 at $4\times10^{-7}$ ($4.9\sigma$). | |
| The profile of the summed rows in the companion's frame (Fig.~\ref{fig:profiles}) is a single line | |
| 5 to 6\,\kms\ wide at half maximum, and nothing beyond 15\,\kms\ of it rises above three times the | |
| profile's scatter. | |
| \begin{table}[t] | |
| \centering | |
| \caption{The secondary's semi-amplitude, with its statistical and systematic uncertainties | |
| (Section~\ref{sec:uncertainty}), the companion's systemic offset with the 16th to 84th percentiles | |
| of 300 draws of whole nights, the summit at an offset of $+1.0$\,\kms\ in units of its nulls' | |
| scatter with its false-alarm probability (Section~\ref{sec:stat}), and the masses at the published | |
| inclinations, $89.639^\circ$, $86.531^\circ$~\cite{Stevenson2026} and | |
| $89.057^\circ$~\cite{Hahlin2026}, against the published values.}\label{tab:k2} | |
| \small\setlength{\tabcolsep}{4pt} | |
| \begin{tabular}{lccc} | |
| \toprule | |
| & TOI-1338 & J0310$-$31 & HD\,4875\\ | |
| \midrule | |
| \tabinput{data/tab_k2.tex} | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \begin{figure}[t] | |
| \centering | |
| \includegraphics[width=\textwidth]{data/maps.png} | |
| \caption{The mean of the cleaned rows along the companion's track over \Ktwo\ and the companion's | |
| systemic offset from the primary's, in units of the map's scatter away from the peak. The dashed | |
| line marks the published \Ktwo, the dotted line zero offset and the cross the peak.}\label{fig:maps} | |
| \end{figure} | |
| \begin{figure}[t] | |
| \centering | |
| \begin{tikzpicture} | |
| \begin{groupplot}[group style={group size=3 by 1, horizontal sep=1.1cm}, width=0.345\textwidth, height=5cm, | |
| xlabel={\Ktwo\ (\kms)}, ymin=-8, ymax=22, grid=major, grid style={gray!20}] | |
| \nextgroupplot[title={TOI-1338 (ESPRESSO)}, ylabel={in units of the nulls' scatter}, xmin=37, xmax=117] | |
| \addplot[thick, no markers] table[x=k2, y=z, col sep=comma]{data/ESPRESSO_J0608-59_k2curve.csv}; | |
| \addplot[gray, dashed] coordinates {(77.16,-8) (77.16,22)}; | |
| \nextgroupplot[title={EBLM J0310$-$31 (HARPS)}, xmin=48, xmax=128] | |
| \addplot[thick, no markers] table[x=k2, y=z, col sep=comma]{data/HARPS_J0310-31_k2curve.csv}; | |
| \addplot[gray, dashed] coordinates {(88.27,-8) (88.27,22)}; | |
| \nextgroupplot[title={HD\,4875 (HARPS)}, xmin=35, xmax=115] | |
| \addplot[thick, no markers] table[x=k2, y=z, col sep=comma]{data/HARPS_J0050-18_k2curve.csv}; | |
| \addplot[gray, dashed] coordinates {(75.13,-8) (75.13,22)}; | |
| \end{groupplot} | |
| \end{tikzpicture} | |
| \caption{The trail over \Ktwo\ at the peak's systemic offset, less the mean of the stricter null's | |
| summits and in units of their scatter, from 300 draws of each null. The dashed line marks the | |
| published \Ktwo.}\label{fig:curves} | |
| \end{figure} | |
| \begin{figure}[t] | |
| \centering | |
| \begin{tikzpicture} | |
| \begin{axis}[width=0.7\textwidth, height=5.5cm, xlabel={offset from the companion's track (\kms)}, | |
| ylabel={mean / scatter}, xmin=-100, xmax=100, ymin=-5, ymax=26, grid=major, grid style={gray!20}, | |
| legend pos=north east, legend style={font=\small, draw=none, fill=white, fill opacity=0.8}] | |
| \addplot[thick, no markers] table[x=offset, y=snr, col sep=comma]{data/ESPRESSO_J0608-59_profile.csv}; | |
| \addlegendentry{TOI-1338} | |
| \addplot[thick, dashed, no markers] table[x=offset, y=snr, col sep=comma]{data/HARPS_J0310-31_profile.csv}; | |
| \addlegendentry{J0310$-$31} | |
| \addplot[thick, dotted, no markers] table[x=offset, y=snr, col sep=comma]{data/HARPS_J0050-18_profile.csv}; | |
| \addlegendentry{HD\,4875} | |
| \end{axis} | |
| \end{tikzpicture} | |
| \caption{The cleaned rows summed in the companion's frame at the peak's \Ktwo\ and systemic | |
| offset, in units of the profile's scatter beyond 30\,\kms.}\label{fig:profiles} | |
| \end{figure} | |
| \subsection{The companion's systemic offset} | |
| The companion's lines sit $+1.05$, $+1.10$ and $+1.25$\,\kms\ from the primary's systemic velocity | |
| in the three systems. The published analyses, which leave the secondary's systemic velocity free | |
| and use the pipelines' M2 line mask, found the same sign and size: $+1.40 \pm 0.06$\,\kms\ in the | |
| ESPRESSO spectra of TOI-1338~\cite{Sebastian2024b} and $+1.20 \pm 0.10$\,\kms\ for | |
| J0310$-$31~\cite{Stevenson2026}. The offset collects the difference between the two templates' | |
| zero points, the primary's G0 template on the scale of its Gaia orbit and GJ\,367's spectrum placed | |
| at its pipeline velocities, and the two stars' differences in gravitational redshift and convective | |
| blueshift, which for J0310$-$31's radii~\cite{Stevenson2026} amount to about 0.1\,\kms\ in | |
| gravitational redshift. Held at zero instead, the offset moves the summit to $77.62$, $88.37$ and | |
| $74.74$\,\kms, by $+0.35$, $+0.25$ and $-0.55$\,\kms, since a constant velocity error of the track | |
| projects onto \Ktwo\ through the phases the spectra sample. At this precision the offset is a | |
| parameter of the fit, and its uncertainty is part of the bootstrap's. | |
| \subsection{Systematic uncertainty} | |
| Each series divided by date, and into alternate nights, gives the semi-amplitudes of | |
| Table~\ref{tab:splits}. The four parts of TOI-1338 span 76.82 to 77.47\,\kms\ and those of | |
| J0310$-$31 87.27 to 88.72\,\kms, giving systematic uncertainties of 0.15 and 0.34\,\kms, below | |
| the published uncertainties of 0.48 and 0.46\,\kms\ that came from splits by observing | |
| season~\cite{Sebastian2024b,Stevenson2026}. HD\,4875's 35 spectra divide into two halves, at | |
| $74.24$ and $75.99$\,\kms, whose standard error of 0.87\,\kms\ is three times the night bootstrap's | |
| 0.30\,\kms\ and makes its total uncertainty 0.93\,\kms. Resampling whole seasons gives smaller | |
| ranges for all three. The systematic term therefore dominates the uncertainty of every | |
| semi-amplitude measured here, as it did in the published analyses. | |
| \begin{table}[t] | |
| \centering | |
| \caption{The semi-amplitude, with the summit's $z$ in brackets, from each part of a series by date | |
| (two parts below 60 spectra) and from alternate nights, each cleaned and measured alone, the range | |
| from resampling whole observing seasons, and the systematic uncertainty, the larger of the parts' | |
| standard error and the seasons' half-range.}\label{tab:splits} | |
| \begin{tabular}{lccc} | |
| \toprule | |
| Subset & TOI-1338 & J0310$-$31 & HD\,4875\\ | |
| \midrule | |
| \tabinput{data/tab_splits.tex} | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \subsection{Robustness and injections} | |
| The five changes of the analysis (Section~\ref{sec:robust}) leave the semi-amplitudes within | |
| 0.05\,\kms\ of the measured value for TOI-1338, within 0.25\,\kms\ for J0310$-$31 and within | |
| 0.40\,\kms\ for HD\,4875 (Table~\ref{tab:robust}), each within its total uncertainty. Each half of | |
| the band finds every companion on its own, at 12 to $20\sigma$ in the units of the nulls' scatter, | |
| and the L\,98-59 template finds it at least as strongly as GJ\,367's. Companions injected at a flux | |
| ratio of $10^{-3}$ a quarter of a period from the real ones are recovered at $-0.15$, $-0.10$ and | |
| $+0.15$\,\kms\ from the injected semi-amplitudes (Table~\ref{tab:inject}), within the | |
| uncertainties and with no sign of a bias from the principal components the cleaning removes. Against the | |
| injections, the real companions' lines match the template's at flux ratios of $1.0$, $0.9$ and | |
| $2.9\times10^{-3}$. | |
| \subsection{Two HARPS series below the threshold} | |
| The 73 archived HARPS spectra of TOI-1338 do not show the companion on their own: the trail's | |
| summit over \Ktwo\ stands at $z = -0.1$, with a false-alarm probability of 0.49. Read at the | |
| published \Ktwo\ and the offset of $+1.0$\,\kms, they give $2.8\sigma$, with 21 of 2000 null draws | |
| as high, as the published analysis of HARPS spectra gave about $3\sigma$~\cite{Sebastian2024b}. For | |
| J1928$-$38, whose companion of $0.276$\,\Msun\ was measured at a signal-to-noise ratio of 3.65 in | |
| 73 HARPS spectra~\cite{Stevenson2026}, the 74 archived HARPS spectra give no summit above the nulls | |
| and $0.3\sigma$ at the published \Ktwo, with 718 of 2000 draws as high. At its brightness, $V = | |
| 11.2$, and its companion's mass, this method does not reach J1928$-$38's secondary in HARPS | |
| spectra. | |
| \section{The southern EBLM binaries}\label{sec:survey} | |
| The same analysis was applied to the 37 EBLM binaries with archived HARPS series, with the | |
| companion's systemic offset held at $+1.0$\,\kms, the value the three systems of | |
| Section~\ref{sec:validation} share, so that each search runs over \Ktwo\ alone. Two series cannot | |
| be searched: J2126$-$28 has 17 spectra above the signal-to-noise cut and J0543$-$57 has 21 spectra | |
| on six nights. Table~\ref{tab:survey} lists the other 35 with each summit's $z$ and false-alarm | |
| probability. | |
| \subsection{Calibration across the survey} | |
| The nulls of the 35 binaries (Fig.~\ref{fig:nulls}) follow the Gumbel law to the depth $10^4$ | |
| draws reach. The median of the 70 fitted shapes is $\xi = -0.02$, nine exceed $+0.1$ and the | |
| largest is $+0.81$, while the seven systems discussed here have $\xi$ between $-0.34$ and $+0.10$ | |
| (Table~\ref{tab:calib}). The survey's highest null, the largest of the 35 standardised summits of | |
| each draw, has a mean of 2.5 and 2.7 and a largest value in $10^4$ draws of 5.1 and 8.5 under the | |
| two nulls. Four binaries reach $5\sigma$ on their own nulls: J0310$-$31 and HD\,4875, J2101$-$45 | |
| at $4.3\times10^{-9}$ ($5.8\sigma$), and TOI-1338's ESPRESSO series outside the survey. Across the | |
| survey J2101$-$45 keeps $1.5\times10^{-7}$ ($5.1\sigma$), HD\,4875 $4\times10^{-7}$ and | |
| J0310$-$31 $3\times10^{-10}$. Three more stand between 3 and $5\sigma$ on their own nulls, | |
| J0649$-$27, J0825$-$67 and J1141$-$37 at 3.9, 3.6 and $3.3\sigma$, but at 3.0, 2.6 and $2.1\sigma$ | |
| across the survey. The next highest, J0057$-$21, J0415$-$16 and J0621$-$50, have probabilities of | |
| 0.05, 0.08 and 0.37 across the survey, the last at the \Ktwo\ of $109.9$\,\kms\ that its | |
| model-based masses predict~\cite{Triaud2017}. | |
| \begin{table}[t] | |
| \centering | |
| \caption{The false-alarm probabilities of the seven companions: the summit at an offset of | |
| $+1.0$\,\kms\ in units of its nulls' scatter ($z$), the highest of the adopted null's $10^4$ draws | |
| in the same units, the Gumbel law's probability and its Gaussian significance, the same with the offset | |
| free within 5\,\kms\ of zero, the probability across the survey's 35 binaries, and the fitted | |
| tail shape under the two nulls. No null draw reaches any of the seven summits, so their empirical | |
| probabilities are at the floor of $10^{-4}$.}\label{tab:calib} | |
| \footnotesize\setlength{\tabcolsep}{3pt} | |
| \begin{tabular}{lcccccc} | |
| \toprule | |
| System & $z$ & Highest null & $p_\mathrm{G}$ ($\sigma$) & Offset free & Across 35 & $\xi$ (phase, nights)\\ | |
| \midrule | |
| \tabinput{data/tab_calib.tex} | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \begin{figure}[t] | |
| \centering | |
| \includegraphics[width=\textwidth]{data/nulls.png} | |
| \caption{Left: the fraction of each binary's $2\times10^4$ null summits at or above a given height | |
| in units of their scatter, under both nulls, for the 35 binaries searched, against the Gumbel law | |
| of unit scatter (dashed), with the six companions of the survey at their observed $z$ and | |
| $p_\mathrm{G}$. Right: the same for the survey's highest null, the largest of the 35 standardised | |
| null summits in each draw, with the Gumbel laws fitted to it (dashed) and the six companions at | |
| their probabilities across the survey.}\label{fig:nulls} | |
| \end{figure} | |
| \subsection{EBLM J2101$-$45} | |
| J2101$-$45's 75 spectra over 2298\,d give \Ktwo\ $= 65.41 \pm 0.32 \pm 0.32$\,\kms\ at an offset of | |
| $+1.45$\,\kms\ (Table~\ref{tab:new}), the systematic term from its eight observing seasons. The | |
| companion holds under every change of the analysis (Table~\ref{tab:robust}): the five variants | |
| give 65.26 to 65.91\,\kms, within 1.1 times the total uncertainty of the measured value, the red | |
| half of the band finds it at 21 and the blue half at 8 times the nulls' scatter, and the L\,98-59 | |
| template at 19. Three of its four parts by date find it on their own, at 6.5 to $9.8$ in the | |
| nulls' units, at 65.41 to 66.16\,\kms; the fourth, at 1.1, does not. A companion injected at a flux | |
| ratio of $10^{-3}$ is recovered at $65.61$\,\kms, and against it the real companion's lines match | |
| the template's at $3.1\times10^{-3}$; one injected at $3\times10^{-4}$ is not recovered. The masses | |
| are $m_1\sin^3 i = 1.415 \pm 0.024$ and $m_2\sin^3 i = 0.552 \pm 0.006$\,\Msun, and since the | |
| binary eclipses, the primary's Gaia DR3 radius of $2.67$\,R$_\odot$ requires $\sin^3 i > 0.993$, so | |
| the masses themselves are within 0.7 per cent of these. The primary, at $1.42$\,\Msun\ and | |
| $2.67$\,R$_\odot$, lies above the main sequence. The model-based masses of | |
| ref.~\cite{Triaud2017}, $1.29 \pm 0.10$ and $0.523 \pm 0.043$\,\Msun, differ from the dynamical | |
| ones by 1.2 and 0.7 times the combined uncertainties. | |
| \subsection{Three candidates} | |
| J0649$-$27, J0825$-$67 and J1141$-$37 do not reach $5\sigma$ on their own nulls and fall to 2.1 to | |
| $3.0\sigma$ across the survey, so their semi-amplitudes (Table~\ref{tab:new}, | |
| Fig.~\ref{fig:diagnostics}) are those their candidate companions would have. Each has a specific | |
| weakness. One of the $10^4$ draws of J0649$-$27's adopted null reaches 6.7 times the nulls' | |
| scatter, against its summit's 7.4; neither half of its 43 spectra finds the companion on its own, at 1.0 and | |
| 1.7 in the nulls' units; and the red half of the band, where an M dwarf's lines carry most of the | |
| signal, peaks instead at $139.8$\,\kms\ and 2.8, while the blue half finds $120.35$\,\kms\ at 4.4. | |
| J0825$-$67 rests on 27 spectra, too few for its parts or alternate nights to be measured alone, and | |
| its uncertainty comes from its five seasons; companions injected along its moved track are not | |
| recovered at either flux ratio, the summit falling at 94\,\kms\ and 0.7 to 0.9 in the nulls' units, | |
| so its spectra do not test the method's recovery. J1141$-$37's 30 spectra divide into halves at 1.1 | |
| and 3.5 in the nulls' units and alternate nights at 5.5 and 0.9, and its seasons give a systematic | |
| uncertainty of 2.6\,\kms. For J0649$-$27 and J1141$-$37 the injections at $10^{-3}$ are recovered | |
| within 0.15\,\kms, and against them the candidate companions' lines would match the template's at | |
| $0.7\times10^{-3}$, the order expected for an M3 dwarf beside an F star. The model-based masses of | |
| ref.~\cite{Triaud2017} for J0649$-$27, $1.45 \pm 0.13$ and $0.371 \pm 0.061$\,\Msun, and for | |
| J1141$-$37, $1.22 \pm 0.10$ and $0.354 \pm 0.050$\,\Msun, are consistent with the minimum masses | |
| the candidates would give, $1.230 \pm 0.071$ and $0.335 \pm 0.012$, and $1.320 \pm 0.090$ and | |
| $0.371 \pm 0.015$\,\Msun, which the eclipses bound by $\sin^3 i > 0.96$. More spectra of these | |
| three would decide them. | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{The 35 searchable EBLM binaries with archived HARPS series: the spectra (nights), the | |
| span, the primary's orbit fitted to the measured velocities, and the summit of the trail over \Ktwo\ | |
| at a systemic offset of $+1.0$\,\kms\ with its $z$ and false-alarm probability. Companions at | |
| $5\sigma$ or more are in bold and candidates between 3 and $5\sigma$ in italics.}\label{tab:survey} | |
| \small | |
| \begin{tabular}{lrrrrrrrr} | |
| \toprule | |
| EBLM & Spectra & Span (d) & $P$ (d) & \Kone\ (\kms) & $e$ & \Ktwo\ (\kms) & $z$ & $p_\mathrm{G}$\\ | |
| \midrule | |
| \tabinput{data/tab_survey.tex} | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{J2101$-$45 and the three candidates, measured with the offset free: the semi-amplitude | |
| with its statistical and systematic uncertainties, the offset with the 16th to 84th percentiles of | |
| 300 draws of whole nights, the false-alarm probability with its Gaussian significance on the | |
| binary's own nulls and across the survey, and the minimum masses with the total uncertainty of | |
| \Ktwo\ carried through.}\label{tab:new} | |
| \footnotesize\setlength{\tabcolsep}{2pt} | |
| \resizebox{\textwidth}{!}{% | |
| \begin{tabular}{lcccccc} | |
| \toprule | |
| EBLM & \Ktwo\ (\kms) & \dvsys\ (\kms) & $p_\mathrm{G}$ ($\sigma$) & Across 35 ($\sigma$) & $m_1\sin^3 i$ (\Msun) & $m_2\sin^3 i$ (\Msun)\\ | |
| \midrule | |
| \tabinput{data/tab_new.tex} | |
| \bottomrule | |
| \end{tabular}} | |
| \end{table} | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{The semi-amplitude, with the summit's $z$ in brackets, as measured and under the five | |
| changes of the analysis of Section~\ref{sec:robust}: three and eight principal components, the | |
| template kept blueward or redward of 5500\,\AA, and the template from L\,98-59's spectra.}\label{tab:robust} | |
| \footnotesize\setlength{\tabcolsep}{3pt} | |
| \begin{tabular}{lcccccc} | |
| \toprule | |
| System & As measured & 3 components & 8 components & 3800--5500\,\AA & 5500--7880\,\AA & L\,98-59\\ | |
| \midrule | |
| \tabinput{data/tab_robust.tex} | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{Companions injected into the spectra a quarter of a period from the real ones: the | |
| flux ratio and \Ktwo\ injected, the semi-amplitude and offset recovered along the moved track with | |
| the summit's $z$, and, where the injection at $10^{-3}$ is recovered, the flux ratio at which the | |
| real companion's lines match the template's ($10^{-3}$).}\label{tab:inject} | |
| \small | |
| \begin{tabular}{lcccccc} | |
| \toprule | |
| System & Flux ratio & \Ktwo\ injected & \Ktwo\ recovered & \dvsys\ (\kms) & $z$ & Real companion\\ | |
| \midrule | |
| \tabinput{data/tab_inject.tex} | |
| \bottomrule | |
| \end{tabular} | |
| \end{table} | |
| \begin{figure}[!htbp] | |
| \centering | |
| \includegraphics[width=\textwidth]{data/diagnostics.png} | |
| \caption{J2101$-$45 and the three candidates. Left: the map over \Ktwo\ and the companion's | |
| systemic offset in units of its scatter, the cross at the peak. Middle: the trail over \Ktwo\ at | |
| the peak's offset in units of the stricter null's scatter, as measured (black) and under the five | |
| changes of the analysis. Right: the cleaned rows summed in the companion's frame at the peak, in | |
| units of the profile's scatter beyond 30\,\kms.}\label{fig:diagnostics} | |
| \end{figure} | |
| \section{Beyond the catalogue orbits}\label{sec:wider} | |
| \subsection{Eight more HARPS binaries} | |
| The selection of Section~\ref{sec:data} needs an orbit from Gaia DR3 or SB9. The ESO archive holds | |
| HARPS series of 41 further EBLM binaries with 15 or more spectra above a signal-to-noise ratio of | |
| 10, and for these the orbit is fitted to the series' own velocities: each spectrum's velocity is | |
| measured against GJ\,367's line depths, a circular-orbit periodogram and a Keplerian fit give the | |
| orbit, the primary's own line depths are made from its spectra at that orbit's velocities and the | |
| velocities measured again against them, twice, and the fit that leaves the least scatter is kept. | |
| The 41 series were searched along their orbits, and the eight in which a companion appears were | |
| built, measured and calibrated as in Section~\ref{sec:method}, the offset held at $+1.0$\,\kms\ | |
| for the calibration. Since the 41 were searched together, each binary's own $p_\mathrm{G}$ is | |
| compounded over them, $1-(1-p_\mathrm{G})^{41}$ (Table~\ref{tab:wider}). | |
| Four companions stand at $5\sigma$ or more across the 41: J2122$-$32, J0425$-$46, J2054$-$32 and | |
| J0535$-$08, at 5.3 to $5.7\sigma$, with \Ktwo\ measured to 0.2 to 0.6 per cent and offsets of | |
| $+0.90$ to $+1.15$\,\kms, as in the systems of Section~\ref{sec:validation}. None has a published | |
| double-lined solution, and Gaia DR3 holds single-lined orbits alone for J2054$-$32 and | |
| J0425$-$46~\cite{GaiaNSS2023}. J0345$-$10, the binary TOI-395, reaches $4.1\sigma$ on its own nulls | |
| and $3.2\sigma$ across the 41, and J1204$-$47, J1328$+$05 and J0610$-$56 reach 3.0 to $3.3\sigma$ | |
| alone; these four remain candidates, J1204$-$47 with a semi-amplitude its NIRPS spectra repeat | |
| (Section~\ref{sec:nir}). | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{The eight companions among the 41 EBLM binaries whose orbits are fitted to their own HARPS | |
| velocities: the spectra (nights), the orbit, the semi-amplitude with its statistical and | |
| systematic uncertainties, the offset with the 16th to 84th percentiles of 300 draws of whole | |
| nights, the false-alarm probability with its Gaussian significance on the binary's own nulls at an | |
| offset of $+1.0$\,\kms\ and compounded over the 41, and the minimum masses. Companions at $5\sigma$ | |
| or more across the 41 are in bold. J1328$+$05's 21 spectra are too few for a systematic | |
| term.}\label{tab:wider} | |
| \footnotesize\setlength{\tabcolsep}{2pt} | |
| \resizebox{\textwidth}{!}{% | |
| \begin{tabular}{lcccccccccc} | |
| \toprule | |
| EBLM & Spectra & $P$ (d) & \Kone\ (\kms) & $e$ & \Ktwo\ (\kms) & \dvsys\ (\kms) & $p_\mathrm{G}$ ($\sigma$) & Across 41 ($\sigma$) & $m_1\sin^3 i$ (\Msun) & $m_2\sin^3 i$ (\Msun)\\ | |
| \midrule | |
| \tabinput{data/tab_wider.tex} | |
| \bottomrule | |
| \end{tabular}} | |
| \end{table} | |
| \subsection{The near infrared}\label{sec:nir} | |
| NIRPS~\cite{Bouchy2017} has observed most of these binaries alongside HARPS since 2023, and in the | |
| near infrared an M dwarf carries a larger share of the light. Twenty of the EBLM binaries with a | |
| HARPS orbit have NIRPS series long enough to search, from the archive's telluric-corrected | |
| products. Three steps change in this band. A solar-type primary has too few lines in it for each | |
| spectrum to be placed on its own, so each is put at the velocity of the HARPS orbit and left | |
| unaligned. Before each residual's cubic is removed, the telluric bands and the samples where the | |
| series' median spectrum falls to 0.2 or below are set aside, since the ratio to a median near zero | |
| would otherwise dominate the fit. The template is the line depths of GJ\,581~\cite{Bonfils2005} from | |
| its 40 best NIRPS spectra, depths above 0.8, which lie at the edges of the telluric bands, removed, | |
| and the template moved by $-2.01$\,\kms\ onto the laboratory wavelengths of the Kurucz line | |
| list~\cite{Kurucz2011}. With this template the three best-measured companions, J1540$-$09's, | |
| J1613$-$76's and J1204$-$47's, sit at offsets of $+3.70$, $+4.45$ and $+0.85$\,\kms, too far apart | |
| to hold one offset, so the probabilities are quoted with the offset free within 10\,\kms\ of zero | |
| and compounded over the 20 series (Table~\ref{tab:nirps}). | |
| J1540$-$09's companion, measured by EBLM~XX at \Ktwo\ $= 60.03 \pm 0.26$\,\kms~\cite{Stevenson2026}, | |
| is recovered from 118 NIRPS spectra at $59.98 \pm 0.50 \pm 0.78$\,\kms\ and $4.0\sigma$. J1928$-$38's | |
| 39 NIRPS spectra put its companion at $62.07 \pm 2.79 \pm 1.53$\,\kms\ against EBLM~XX's $64.35 \pm | |
| 2.82$, at $1.4\sigma$, where its 74 HARPS spectra show nothing (Section~\ref{sec:validation}). | |
| J1613$-$76, whose HARPS series shows no companion, gives one at $70.11 \pm 1.76 \pm 3.02$\,\kms\ and | |
| $3.5\sigma$ from 22 spectra, a candidate, and J1204$-$47's NIRPS spectra repeat its HARPS | |
| semi-amplitude within 0.6\,\kms. J0353$-$16 and J0055$-$00 reach 2.0 and $1.2\sigma$. | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{The NIRPS series in which a companion appears: the spectra (nights), the semi-amplitude | |
| with its statistical and systematic uncertainties, the offset with the 16th to 84th percentiles of | |
| 300 draws of whole nights, the false-alarm probability with the offset free within 10\,\kms\ on the | |
| binary's own nulls and compounded over the 20 series searched, the companion's minimum mass, and | |
| the semi-amplitude measured elsewhere.}\label{tab:nirps} | |
| \footnotesize\setlength{\tabcolsep}{2pt} | |
| \resizebox{\textwidth}{!}{% | |
| \begin{tabular}{lccccccc} | |
| \toprule | |
| EBLM & Spectra & \Ktwo\ (\kms) & \dvsys\ (\kms) & $p_\mathrm{G}$ ($\sigma$) & Across 20 ($\sigma$) & $m_2\sin^3 i$ (\Msun) & \Ktwo\ elsewhere (\kms)\\ | |
| \midrule | |
| \tabinput{data/tab_nirps.tex} | |
| \bottomrule | |
| \end{tabular}} | |
| \end{table} | |
| \subsection{Masses and radii from the eclipses}\label{sec:eclipse} | |
| The binaries eclipse, and TESS~\cite{Ricker2015} light curves, SPOC's at the shortest cadence or | |
| else TESS-SPOC's or QLP's read through Lightkurve~\cite{Lightkurve2018}, give the inclination and | |
| the radii. Both eclipses are modelled where the spectroscopic orbit places them, $e$ and $\omega$ | |
| held at the orbit's: the transit is a dark disk of radius ratio $R_2/R_1$ across a quadratically | |
| limb-darkened primary, parametrised by $q_1$ and $q_2$~\cite{Kipping2013}, diluted by the | |
| companion's share $L_2$ of the TESS-band light, and the occultation removes $L_2$ times the hidden | |
| fraction of the companion's disk, its limb darkening held at 0.4 and 0.3; each exposure is | |
| integrated over its length. The time of periastron, the period, $R_2/R_1$, $a/R_1$, $\cos i$, | |
| $q_1$, $q_2$ and $L_2$ are fitted from several starts, small and grazing, by least squares and | |
| sampled with emcee~\cite{ForemanMackey2013}, the primary's radius held to its TIC | |
| value~\cite{Stassun2019} within 7 per cent except for J0535$-$08 and J0345$-$10. The inclination | |
| turns the minimum masses into masses, and $a/R_1$ with $a\sin i = (K_1+K_2)P\sqrt{1-e^2}/2\pi$ gives | |
| both radii (Table~\ref{tab:eclipse}). | |
| J0535$-$08's eclipses, 18.3 and 2.7 per cent deep at an impact parameter of 0.19, give masses of | |
| $0.970 \pm 0.011$ and $0.521 \pm 0.003$\,\Msun\ and radii of 1.209 and 0.505\,R$_\odot$, both | |
| stars to within one per cent without the TIC radius. In J2122$-$32 the companion passes clear of | |
| the primary at conjunction, its impact parameter 1.76 against $1 + R_2/R_1 = 1.48$, and only its | |
| occultation, 1.3 per cent deep, is seen; J2054$-$32, J0345$-$10 and J1328$+$05 graze, so their | |
| companions' radii stay loose while their masses, with $\sin^3 i$ between 0.976 and 0.997, do not. | |
| J1613$-$76's points scatter 2.5 times their errors about the model, which understates its radius | |
| intervals by that factor. J0425$-$46's primary has left the main sequence, at 1.59\,\Msun\ and | |
| 3.21\,R$_\odot$, beside a companion of 0.743\,\Msun\ and 0.704\,R$_\odot$. | |
| \begin{table}[!htbp] | |
| \centering | |
| \caption{Masses and radii of the binaries of Tables~\ref{tab:wider} and~\ref{tab:nirps} from their | |
| TESS eclipses: the transit and occultation windows observed, the inclination with its 16th to 84th | |
| percentiles, the companion's share of the TESS-band light, the masses with the semi-amplitude's | |
| total uncertainty carried through, and the radii with the eclipse fit's percentiles. J1613$-$76's | |
| \Ktwo\ is its NIRPS value and the others' their HARPS values; the masses of the candidates are | |
| those their companions would have.}\label{tab:eclipse} | |
| \footnotesize\setlength{\tabcolsep}{3pt} | |
| \resizebox{\textwidth}{!}{% | |
| \begin{tabular}{lccccccc} | |
| \toprule | |
| EBLM & Windows & $i$ (deg) & $L_2$ & $M_1$ (\Msun) & $M_2$ (\Msun) & $R_1$ (R$_\odot$) & $R_2$ (R$_\odot$)\\ | |
| \midrule | |
| \tabinput{data/tab_eclipse.tex} | |
| \bottomrule | |
| \end{tabular}} | |
| \end{table} | |
| \section{Discussion}\label{sec:discussion} | |
| The semi-amplitudes measured here differ from the published ones by $-0.15$ to $+0.16$\,\kms, | |
| within the published uncertainties, although the analysis makes its own choice at each step. The | |
| template is an observed M1\,V spectrum rather than a line mask or a synthetic spectrum; the | |
| primary's spectrum is removed by division by the series' median in the primary's frame and by | |
| principal components of the cross-correlation rows, rather than by a singular value decomposition | |
| of the spectra; the significance is measured against nulls built from the same rows rather than | |
| from a fit to the map; and the uncertainty comes from resampling whole nights and dividing the | |
| series by date. The agreement indicates that the published dynamical masses of TOI-1338 and EBLM | |
| J0310$-$31 do not depend on those analysis choices at the level of their quoted uncertainties, and | |
| that the near-infrared semi-amplitude of HD\,4875's companion holds in the optical. | |
| The calibration separates the height of a summit from its probability. In units of the nulls' | |
| scatter the four companions stand 13.8 to 19.4 high, but the nulls are drawn $10^4$ times each, so | |
| the empirical probability stops at $10^{-4}$, and beyond it the probability rests on the Gumbel | |
| law. That law fits the nulls to the depth they reach, and for the four companions the fitted | |
| tails are as light as Gumbel's or lighter, so their probabilities are not optimistic. Their | |
| summits also stand 11 to 16 times the nulls' scatter beyond the highest of the adopted null's | |
| $10^4$ draws. | |
| Across the survey the probability of the weakest of the four, HD\,4875, is $4\times10^{-7}$, | |
| and for J2101$-$45 $1.5\times10^{-7}$. The three candidates show why the survey-wide accounting | |
| matters: at 3.3 to $3.9\sigma$ alone they would each look like detections, and 35 binaries searched | |
| over \Ktwo\ make summits of that height expected. | |
| The archive's own completeness matters as much as the method. It holds 114 HARPS spectra of | |
| J0310$-$31 above a signal-to-noise ratio of 10, against the 71 of the published analysis, and 260 | |
| usable ESPRESSO spectra of TOI-1338, two and a half times the published set. A detection's | |
| significance grows with the square root of the number of spectra at fixed quality, and the | |
| candidates, with 27 to 43 spectra, are where more spectra would count. | |
| The method has the limits of any optical analysis of these binaries. Toward 0.1\,\Msun\ the | |
| companion's optical flux falls to $10^{-4}$ of the primary's and below, and the HARPS series of | |
| J1928$-$38 and of the lowest-mass companions do not reach them; no injection at | |
| $3\times10^{-4}$ is detected, and only J1141$-$37's returns the injected \Ktwo, at 1.8 in the | |
| nulls' units. The near infrared carries a larger share of the | |
| companion's light, and there the method recovers J1540$-$09's published companion and reaches | |
| J1928$-$38's and J1613$-$76's, which their HARPS series do not (Section~\ref{sec:nir}). | |
| \section{Conclusion}\label{sec:conclusion} | |
| An analysis of public ESO archive spectra, independent of the published pipelines, recovers the | |
| M-dwarf companions of TOI-1338, EBLM J0310$-$31 and HD\,4875 with false-alarm probabilities of | |
| $10^{-11}$, $8\times10^{-12}$ and $1\times10^{-8}$ and measures their semi-amplitudes as $77.27 \pm | |
| 0.18$, $88.12 \pm 0.38$ and $75.29 \pm 0.93$\,\kms, within $0.16$\,\kms\ of the published values, | |
| with masses within 0.6 per cent of the published dynamical masses. HD\,4875's companion, measured | |
| before in the near-infrared alone, is detected here in the optical. The companion's lines sit | |
| $+1.05$ to $+1.25$\,\kms\ from the primary's systemic velocity in all three, and leaving that offset | |
| free is needed for accuracy at the tenth of a kilometre per second. Across the 35 searchable | |
| southern EBLM binaries with archived HARPS series, the companion of EBLM J2101$-$45 is measured for | |
| the first time, at a false-alarm probability of $1.5\times10^{-7}$ across the survey, with \Ktwo\ | |
| $= 65.41 \pm 0.46$\,\kms\ and minimum masses of $1.415$ and $0.552$\,\Msun; the companions of | |
| J0649$-$27, J0825$-$67 and J1141$-$37 remain candidates. Among 41 EBLM binaries outside the | |
| catalogue orbits, the companions of J2122$-$32, J0425$-$46, J2054$-$32 and J0535$-$08 are measured | |
| for the first time at 5.3 to $5.7\sigma$ across the 41, and with their TESS eclipses their masses | |
| and radii follow without stellar models; NIRPS spectra recover J1540$-$09's companion within | |
| 0.05\,\kms\ of its published semi-amplitude. | |
| \section*{Acknowledgements} | |
| Based on data obtained from the ESO Science Archive Facility, from ESO programmes | |
| 092.D-0261(A), 095.C-0367(A), 099.C-0138(A), 099.C-0138(B), 0101.C-0510(C), 103.2024.001, | |
| 106.212H.001, 106.212H.007, 106.212H.009, 106.212H.010, 106.216B.001, 106.216B.002, 106.216B.003, | |
| 106.21TJ.001, 1101.C-0721(A), 112.25GZ.001, 112.25GZ.002, 112.25GZ.003, 112.25GZ.004, | |
| 112.25KP.001, 112.25UD.001, 113.26NT.001, 115.27V0.001, 115.27V0.002, 116.28LH.001, 116.28LL.001 | |
| and 116.28LL.002. | |
| This work has made use of data from the European Space Agency mission Gaia, processed by the Gaia | |
| Data Processing and Analysis Consortium, funded by national institutions, in particular the | |
| institutions participating in the Gaia Multilateral Agreement; of the SIMBAD database and the | |
| VizieR catalogue access tool, operated at CDS, Strasbourg, France; and of the SB9 catalogue of | |
| spectroscopic binary orbits. | |
| \section*{Data availability} | |
| The spectra are public in the ESO Science Archive. The analysis is the binaries approach of the | |
| seti-tools package that holds this paper (\texttt{seti.py composition binaries}, whose modes build | |
| the rows, search, calibrate and measure), and its coverage store records the run as | |
| \texttt{binaries\_eblm\_known\_orbits}. The paper's data folder holds the measurements and their | |
| maps, curves and profiles, the robustness and injection measurements, the calibrations, the | |
| survey's search results, the eclipse fits, and the tables written from them. | |
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