Tidal Parameters Research Articles

A new high-precision gravity solid tidal model for Precision Gravity Measurement Facility

It is significant for establishing gravity datum to construct precise gravity solid tidal model. A simple method with relatively low performance is to interpolate tidal parameters from the global gravity solid tide models. A competitive approach is to determine local gravity solid tidal model by harmonic analysis using long-time serial gravity observations. In this paper a new high-precision gravity solid tidal model for Precision Gravity Measurement Facility is estimated from two co-site gravimeters in the cave laboratory using modern international standard data processing techniques, whose accuracy is evaluated further by comparing with previous publications. The results show that: (1) the determined gravity solid tidal models from two co-site gravimeters are in good agreement with each other, of which the maximum differences for amplitude factors and phase delays don't exceed 0.01700% and 2.50990°, respectively. (2) the performance of the obtained gravity solid tidal model is 0.00411 for amplitude factors and 0.24120° for phase delays, which is a little better than that of previous publications using superconducting gravity data from Wuhan station. (3) our results and methods are corrective and effective. (4) our model is tiny different from that provided by Wuhan station, which implies that it is necessary to construct a gravity solid tidal model for Precision Gravity Measurement Facility, rather than just adopting existing models at Wuhan station. Our results are helpful in realizing the goal of Precision Gravity Measurement Facility.

Preliminary assessment of a tidal test site on the Minho estuary

Abstract A characterization of tidal parameters at estuaries and the potential of these to be used for tidal energy extraction is performed. The tidal parameters analysed are crossed with estuaries around the world to obtain a selection of potentially feasible locations. A case study is addressed regarding the preliminary evaluation of location and overall characteristics of a tidal test centre and tidal farm in the Minho River estuary at the border between Spain and Portugal. Technical and environmental requirements are analysed to evaluate the potential of the location. Infrastructure characteristics, electrical connections and other equipment specification are determined considering the potential capacity available in the estuary; alternative configurations are studied and compared. Based on the study, the maximum capacity of the tidal technology centre is taken into account to produce an initial estimate of costs to deploy a device. The feasibility study shows that the economic evaluation of the investment is favourable, and the development of this tidal test site is feasible and economically viable. Even, a private organism could develop this project without any type of direct public investment.

The Huelva (Spain) Tsunami-Ready Station and Its Interaction with Storm Emma (March 2018)

The sea level station operating since 1996 at Mazagón (Huelva, Spain) has been progressively upgraded to fit tsunami warning requirements, due to its location in one of the main regions at risk. Its radar water level sensor was complemented in 2017, with the addition of a pressure sensor. The performance of both sea level sensors and their response to sea level oscillations, at different frequencies, is assessed. Particular emphasis is put on the effect of extreme events, such as Storm Emma, when alternative methods to obtain 1-min data are tested, in contrast to the one based on arithmetic means. The overall differences are small, for the whole period of study (centered-root-mean-square-error below 1 cm, for 5-min, and hourly data; similar tidal parameters and sea level oscillations with periods between 30 s and 5 min). However, during Storm Emma, the pressure sensor presents sensibly lower readings than the radar, with the centered-root-mean-square-error rising to 80 mm on the March 2nd 2018. A new method to compute 1-min data, based on medians, reduced this value to 10 mm for the same day.

Efficient tidal dissipation in Deimos

After its formation Deimos, Mars' second and outer satellite, is expected to have been catastrophically disrupted, possibly multiple times, by planetesimals left over from primary terrestrial planet accretion. The expected last time such a disruption occurred is ca. 4.3 Gyr ago. After this disruption and its subsequent expected coagulation from a thin circum-martian ring, Deimos likely experienced a gradual excitation in its orbital eccentricity from flybys with other leftover planetesimals. The cumulative effect from these encounters indicates that Deimos' eccentricity should have been up to an order of magnitude greater than it is now, with a mean value of 1.5−1.1+2.3×10−3. Based on this eccentricity excitation, Deimos' calculated age, the assumption of synchronous rotation throughout its existence and that the eccentricity is only tidally damped in Deimos and not in Mars, the resulting tidal parameters of Deimos are k2D/QD=3.52−2.89+2.04 × 10−5 (2σ). The large uncertainties in the tidal parameters are caused by the wide eccentricity distribution resulting from encounters with stray planetesimals. Employing a simple tidal model, I calculate that Deimos' shear viscosity is ηD=2.13−0.79+9.77×1012 Pa s, far lower than that of cold rock. Under the assumption that Deimos' interior is akin to a rubble pile its internal rigidity is μrp ~ 120 MPa, while the tidal Love number falls in the range k2D ~ (1 − 7) × 10−5. The low rigidity and viscosity indicate that most of the tidal damping likely occurs in the surface regolith layer. As a consequence, Deimos appears to be near its maximum tidal dissipation efficiency (QD ~ 1). An early episode of chaotic tumbling could have initially lowered Deimos' eccentricity to near zero, but this was likely followed by an excitation of its eccentricity through a past resonance with Phobos ca. 2 Gyr ago. This resonance crossing excited Deimos' eccentricity to 0.002, and subsequent synchronous damping over the next 2.5 Gyr requires k2D/QD ~ 7 × 10−5, which still implies efficient damping in Deimos.

Tidal Breathing Parameters Measured by Structured Light Plethysmography in Newborns: Is It Feasible in Neonatal Intensive Care Unit?

Structured light plethysmography (SLP) is a novel and noncontact respiratory assessment technique. It provides tidal breathing measurement in patients difficult to cooperate. In this study, we aimed to determine data for tidal breathing parameters measured by SLP in newborns. Infants between 2 and 5 days of life without having any respiratory symptoms were eligible for this observational study. In total, 5 minutes of tidal breathing was recorded using SLP (Thora-3Di, PneumaCare Ltd, Cambridge, U.K.) in each infant. Various tidal breathing parameters including timing indices, flow-based parameters, and regional parameters were obtained from SLP data. A total of 57 infants underwent measurements in the study. Evaluable recordings from 42 term and 11 late preterm infants were analyzed. Median gestational age and birthweight of the infants were 38 (37-39) weeks and 3,195 (2,790-3,585) g, respectively. In terms of flow-based parameters, "tidal inspiratory flow at 50% of inspiratory volume divided by tidal expiratory flow at 50% of expiratory volume" was 1.29 (1.13-1.53). Relative contribution of the thorax to each breath in percentage was measured as 38.67 (28.21-43.60). Median values of left-right hemithoracic asynchrony and thoraco-abdominal asynchrony were 6.92 (5.35-9.04) and 17.96 (12.98-36.44) degrees in the study population, respectively. There were no differences in tidal breathing parameters except "hemithoracic asynchrony" between term and late preterm infants. Hemithoracic asynchrony was significantly lower in term neonates than late preterms. SLP was found to be feasible to obtain measures of tidal breathing parameters in newborns and it could be performed successfully even in the first days of life.

Exoplanet system Kepler-2 with comparisons to Kepler-1 and 13

We have carried out an intensive study of photometric (Kepler Mission) and spectroscopic data on the system Kepler-2 (HAT-P-7A) using the dedicated software WinFitter. The mean individual data-point error of the normalized flux values for this system is 0.00015, leading to the model's specification for the mean reference flux to an accuracy of $\sim$0.5 ppm. This testifies to the remarkably high accuracy of the binned data-set, derived from over 1.8 million individual observations. Spectroscopic data are reported with the similarly high-accuracy radial velocity amplitude measure of $\sim$2 m s$^{-1}$. The analysis includes discussion of the fitting quality and model adequacy. Our derived absolute parameters for Kepler-2 are as follows: $M_p$ (Jupiter) 1.80 $\pm$ 0.13; $R_{\star}$ 1.46 $\pm 0.08 \times 10^6$ km; $R_p$ 1.15 $\pm 0.07 \times 10^5$ km. These values imply somewhat larger and less condensed bodies than previously catalogued, but within reasonable error estimates of such literature parameters. We find also tidal, reflection and Doppler effect parameters, showing that the optimal model specification differs slightly from a `cleaned' model that reduces the standard deviation of the $\sim$3600 binned light curve points to less than 0.9 ppm. We consider these slight differences, making comparisons with the hot-jupiter systems Kepler-1 (TrES-2) and 13.

WITHDRAWN: Sea level rise and estuarine tidal dynamics: A review

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Natural dynamics overshadow anthropogenic impact on marine fauna at an urbanised coastal embayment

Understanding vulnerabilities of coastal ecosystems facing anthropogenic use is a precondition for management decisions and development planning. This can be challenging in urbanised areas with multiple activities affecting different faunal communities. The aim of this study was to provide a holistic understanding of the relative importance of anthropogenic and natural variables for macroinfauna, epifauna and fish in a heavily modified waterbody (HMWB) designated under the EU Water Framework Directive (WFD). The study area, Swansea Bay (Wales, UK), had two regularly dredged industrial ports, three estuaries, a wastewater discharge point and a dredge-spoil disposal site. Wave and tidal current models were constructed, and environmental data were gathered by field studies. Biota were assessed by grab sampling and dredging. Modelled and empirical data were combined in a Distance-based Linear Model (DistLM) that quantified how much of the faunal variation was explained by wave exposure and tidal currents, sediment characteristics and other environmental factors, and by anthropogenic usage. Wave and tidal current parameters explained over 50% of the variation in all biota. Infauna communities were further linked with sediment properties and epibenthos with distance to estuaries. Fish and epibenthos were affected by a dredge-spoil disposal site, but none of the faunal communities was affected by the wastewater outfall. Biota were predominantly driven by the natural hydrodynamic regime while anthropogenic factors had secondary influence. The study highlighted that ecosystems driven by a strong hydrodynamic regime can be relatively resistant to human activities.

Tidal Parameters as a Tool for the Determination of the Coordinates of the SLR Stations

Abstract One of the primary objectives of satellite geodesy is the determination of coordinates of the satellite laser ranging (SLR) stations. This task is conducted by using laser ranging techniques. The main goal of the current study was to assess the influence of using varied values of the tidal parameters (Love h 2 and Shida l 2 numbers) on the determination of the positions of chosen SLR stations. The obtained results are presented for coordinates determination conducted for six SLR stations: Mt Stromlo (no. 7825, Australia), Matera (no. 7941, Italy), Grasse (no. 7845, France), McDonald (no. 7080, USA), Arequipa (no. 7403, Peru) and Beijing (no. 7249, China). The analysis covers SLR data for 2 satellites (LAGEOS1 and LAGEOS2), which were observed for 10 consecutive years (from 2008 to 2018). The analysis was performed using the ITRF2014 reference frame in two scenarios of calculations. In scenario 1, the SLR stations coordinates were calculated using the nominal values as per the International Earth Rotation and Reference System Service (IERS) standards recommendation of the Love/Shida numbers: h 2 = 0.6078, l 2 = 0.0847. In scenario 2, the coordinates were estimated using the harnessing values of the Love/Shida numbers (h 2 = 0.6140 and l 2 = 0.0876), which were proposed by authors in a previous publication. The effect of the application of different values of the Love/Shida numbers for the determination of SLR stations coordinates was scrutinized.

Amplitude Delta Factors and Phase Shifts of Tidal Waves for the Earth with the Ocean on the Territory of Russia

The results of the calculation of the amplitude delta factors and phase shifts of semidiurnal and diurnal tidal waves for a rotating self-gravitating ellipsoidal inelastic Earth with the ocean are presented, and the method used to calculate them is briefly described. The maps of distribution of the considered tidal parameters are constructed for the M2 and K1 waves over the territory of Russia; the zones with extreme and generally accepted values are identified. A discussion of the revealed features of the distribution of the amplitude factors and phase shifts on the territory of the Russian Federation is presented.

Measuring the neutron star equation of state with gravitational waves: The first forty binary neutron star merger observations

Gravitational waves from binary neutron star coalescences contain rich information about matter at supranuclear densities encoded by the neutron star equation of state. We can measure the equation of state by analyzing the tidal interactions between neutron stars, which is quantified by the tidal deformability. Multiple merger events are required to probe the equation of state over a range of neutron star masses. The more events included in the analysis, the stronger the constraints on the equation of state. In this paper, we build on previous work to explore the constraints that LIGO and Virgo are likely to place on the neutron star equation of state by combining the first 40 binary neutron star detections, a milestone we project to be reached during the first year of accumulated design-sensitivity data. We carry out Bayesian inference on a realistic mock dataset of binaries to obtain posterior distributions for neutron star tidal parameters. In order to combine posterior samples from multiple observations, we employ a random forest regressor, which allows us to efficiently interpolate the likelihood distribution. Assuming a merger rate of $1540\text{ }\text{ }{\mathrm{Gpc}}^{\ensuremath{-}3}\text{ }\text{ }{\mathrm{yr}}^{\ensuremath{-}1}$ and a LIGO-Virgo detector network operating for 1 year at the sensitivity of the third-observation run, plus an additional 8 months of design sensitivity, we find that the radius of a $1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ neutron star can be constrained to better than $\ensuremath{\sim}10%$ at 90% confidence. The pressure at twice the nuclear saturation density can be constrained to better than $\ensuremath{\sim}45%$ at 90% confidence. We show that these results are robust to our choice of parametrization for the neutron star equation of state.

The multiparametric method of analyzing the lunar dynamic processes

The present paper focuses on studying the lunar dynamic processes. As selenophysics is a complex system and moonquakes are complex multi-parametric systems too, the analysis of moonquakes and the development of an analytical theory of such processes require the application of robust methods and using multiparametric calculations. Moreover, selenophysics is a more complex system than geophysics. In this connection, the study of lunar processes and determination of moonquakes parameters require reliable estimates of the results obtained and application of the methods of complex system physics. The observations from “Apollo” space mission were used in the study, and the multiparametric correlation method was developed for their processing. Currently, one uses various methods, similar to the ones for the Earth’s seismic process investigation, such as seismic interferometry for deep moonquakes, time scales analysis, seismic phenomena magnitude gradient change, solution of the inverse problem of signals reflection on “Apollo” stations. Now, on the basis of moonquakes data, the internal structure of the Moon is being studied. Using the moonquakes data, a model of the lunar tidal parameters had been developed and was later refined by “GRAIL” (gravitation), “LRO” (shape), and “LLR” (rotation) space missions’ data. As a result, in the areas where deep moonquakes occurred, the inner layer of the Moon with low viscosity was found. In the present work, the author’s method of analyzing moonquakes allowing to conduct multiparametric analysis of seismic time series observations was applied. The method was developed for the investigation of seismic processes occurring on the Earth using the space observations. In order to provide qualitative description of moonquakes dynamical parameters, the special software was developed.

Model of Sedimentation and Bed-Load Concentration Patterns in Sanctuary of Porong River of Sidoarjo Post Mudflow

This paper shows the distribution patterns and the sediment concentrations as well as the models of the sedimentation process to bathymetric profile changes (Bed-changes). This research aims to analyze the hydro-oceanographic conditions in order to make considerations to empower the impacts of Coastal Management Planning of the mudflow in Porong Sidoarjo. The model has been a sediment transport suspension and bed load with tidal parameters and sedimentation data. This model has resulted in the sedimentation distribution, which has been more or less controlled by tidal activity. This model has also resulted in changes to the bathymetry profile for 360 hours, then estimated for the next period of 1, 5, and 10 years for the future. Changes in the seabed rose from 0.50 to 0.87 meters over the next 10 years in the research site. Model errors have been based on the flow velocity in the range of 2.4% to 6.1%.

Constraints on Microscopic and Phenomenological Equations of State of Dense Matter from GW170817

We discuss the constraints on the equation of state (EOS) of neutron star matter obtained by the data analysis of the neutron star-neutron star merger in the event GW170807. To this scope, we consider two recent microscopic EOS models computed starting from two-body and three-body nuclear interactions derived using chiral perturbation theory. For comparison, we also use three representative phenomenological EOS models derived within the relativistic mean field approach. For each model, we determine the β -stable EOS and then the corresponding neutron star structure by solving the equations of hydrostatic equilibrium in general relativity. In addition, we calculate the tidal deformability parameters for the two neutron stars and discuss the results of our calculations in connection with the constraints obtained from the gravitational wave signal in GW170817. We find that the tidal deformabilities and radii for the binary’s component neutron stars in GW170817, calculated using a recent microscopic EOS model proposed by the present authors, are in very good agreement with those derived by gravitational waves data.

Tidal response of rocky and ice-rich exoplanets

The amount of detected planets with sizes comparable to that of the Earth is increasing drastically. Most of the Earth-size planet candidates orbit at close distances from their central star, and therefore are subjected to large tidal forces. Accurate determination of the tidal parameters of exoplanets taking into account their interior structure and rheology is essential to better constrain their rotational and orbital history, and hence their impact on climate stability and planetary habitability. In the present study, we compute the tidal response of rocky and ice-rich solid exoplanets for masses ranging between 0.1 and 10 Earth masses using a multilayer approach and an Andrade rheology. We show that the amplitude of tidal response, characterized by the gravitational Love number,k2, is mostly controlled by self-gravitation and increases as a function of planet mass. For rocky planets,k2depends mostly on the relative size of the iron core, and hence on the bulk iron fraction. For ice-rich planets, the presence of outer ice layers reduces the amplitude of tidal response compared to ice-free rocky planets of similar masses. For both types of planet (rocky and ice-rich), we propose relatively simple scaling laws to predict the potential Love number value as a function of radius, planet mass and composition. For the dissipation rate, characterized by theQ−1factor, we did not find any direct control by the planet mass. The dissipation rate is mostly sensitive to the forcing frequency and to the internal viscosity, which depends on the thermal evolution of the planet, which is in turn controlled by the planet mass and composition. The methodology described in the present study can be applied to any kind of solid planet and can be easily implemented into any thermal and orbital evolution code.

Determination of the local tidal parameters for the Borowiec station using Satellite Laser Ranging data

The values of regional tidal parameters h2, l2 associated with the tidal variations of ground stations were estimated for the Polish Satellite Laser Ranging (SLR) station Borowiec using SLR data. The study is based on satellite observations taken by the global network of ground stations during the period from January 1, 1999 until January 1, 2019 for monthly orbital arcs of the satellites LAGEOS-1 and LAGEOS-2. The adjusted regional values for h2 equalling 0.7308 ± 0.0008 and l2 equalling 0.1226 ± 0.0003 are discussed and compared with the nominal values of h2 and l2 given in the the International Earth Rotation and Reference Systems Service (IERS) standards and with other estimations of these parameters. Furthermore, the influence of the tidal parameters changes on estimation of the Borowiec station coordinates in the ITRF2014 reference frame was investigated. The analysis was carried out in two variants. The first one consisted in the determination of the Borowiec station coordinates with the use of the nominal values of the tidal parameters: h2 = 0.6078 and l2 = 0.0847 (IERS recommended values). In the second one, the Borowiec station coordinates were determined using the local tidal parameters estimated in this paper (h2 = 0.7308 ± 0.0008 and l2 = 0.1226 ± 0.0003). The differences between X, Y ,Z for Variant 1 and Variant 2 are −3.5, 3.3 and 4.2 mm, respectively.

Investigating the clinical use of structured light plethysmography to assess lung function in children with neuromuscular disorders.

BackgroundChildren and young people with neuromuscular disorders (NMD), such as Duchenne Muscular Dystrophy (DMD), develop progressive respiratory muscles weakness and pulmonary restriction. Pulmonary function monitoring of the decline in lung function allows for timely intervention with cough assist techniques and nocturnal non-invasive ventilation (NIV). NMD may find the measurement of lung function difficult using current techniques. Structured Light Plethysmography (SLP) has been proposed as a novel, non-contact, self-calibrating, non-invasive method of assessing lung function. The overarching aim of this study was to investigate the use of SLP as a novel method for monitoring respiratory function in children with neuromuscular disease.MethodsSLP thoraco-abdominal (TA) displacement was correlated with forced vital capacity measurements recorded by spirometry and the repeatability of the measurements with both methods examined. SLP tidal breathing parameters were investigated to assess the range and repeatability of regional right and left side TA displacement and rib cage and abdominal wall displacement.ResultsThe comparison of the FVC measured with SLP and with spirometry, while having good correlation (R = 0.78) had poor measurement agreement (95% limits of agreement: -1.2 to 1.2L) The mean relative contribution of right and left TA displacement in healthy controls was 50:50 with a narrow range. Repeatability of this measure with SLP was found to be good in healthy controls and moderate in NMD children with/without scoliosis but with a wider range. The majority of the control group displayed a predominant rib cage displacement during tidal breathing and those who displayed predominant abdominal wall displacement showed displacement of both regions close to 50:50 with similar results for the rib cage and abdomen. In comparison, children with NMD have a more variable contribution for all of these parameters. In addition, SLP was able to detect a reduction in abdominal contribution to TA displacement with age in the DMD group and detect paradoxical breathing in children with NMD. Using SLP tracings during tidal breathing we were able to identify three specific patterns of breathing amongst healthy individuals and in children with NMD.ConclusionsSLP is a novel method for measuring lung function that requires limited patient cooperation and may be especially useful in children with neuromuscular disorders. Measuring the relative contributions of the right and left chest wall and chest versus abdominal movements allows a more detailed assessment.

Rotation Period Evolution in Low-mass Binary Stars: The Impact of Tidal Torques and Magnetic Braking

Abstract We examine how tides, stellar evolution, and magnetic braking shape the rotation period (P rot) evolution of low-mass stellar binaries up to orbital periods (P orb) of 100 days across a wide range of tidal dissipation parameters using two common equilibrium tidal models. We find that many binaries with P orb ≲ 20 days tidally lock, and most with P orb ≲ 4 days tidally lock into synchronous rotation on circularized orbits. At short P orb, tidal torques produce a population of fast rotators that single-star-only models of magnetic braking fail to produce. In many cases, we show that the competition between magnetic braking and tides produces a population of subsynchronous rotators that persists for 1 Gyr, even in short-P orb binaries, qualitatively reproducing the subsynchronous eclipsing binaries discovered in the Kepler field by Lurie et al. Both equilibrium tidal models predict that binaries can tidally interact out to P orb ≈ 80 days, while the constant phase lag tidal model predicts that binaries can tidally lock out to P orb ≈ 100 days. Tidal torques often force the P rot evolution of stellar binaries to depart from the long-term magnetic-braking-driven spin-down experienced by single stars, revealing that P rot is not a valid proxy for age in all cases, i.e., gyrochronology can underpredict ages by up to 300% unless one accounts for binarity. We suggest that accurate determinations of orbital eccentricties and P rot can be used to discriminate between which equilibrium tidal models best describe tidal interactions in low-mass binary stars.

Surrogate model for an aligned-spin effective-one-body waveform model of binary neutron star inspirals using Gaussian process regression

Fast and accurate waveform models are necessary for measuring the properties of inspiraling binary neutron star systems such as GW170817. We present a frequency-domain surrogate version of the aligned-spin binary neutron star waveform model using the effective-one-body formalism known as SEOBNRv4T. This model includes the quadrupolar and octopolar adiabatic and dynamical tides. The version presented here is improved by the inclusion of the spin-induced quadrupole moment effect and completed by a prescription for tapering the end of the waveform to qualitatively reproduce numerical relativity simulations. The resulting model has 14 intrinsic parameters. We reduce its dimensionality by using universal relations that approximate all matter effects in terms of the leading quadrupolar tidal parameters. The implementation of the time-domain model can take up to an hour to evaluate using a starting frequency of 20 Hz, and this is too slow for many parameter estimation codes that require $\mathcal{O}({10}^{7})$ sequential waveform evaluations. We therefore construct a fast and faithful frequency-domain surrogate of this model using Gaussian process regression. The resulting surrogate has a maximum mismatch of $4.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ for the Advanced LIGO detector and requires $\ensuremath{\sim}0.13\text{ }\text{ }\mathrm{s}$ to evaluate for a waveform with a starting frequency of 20 Hz. Finally, we perform an end-to-end test of the surrogate with a set of parameter estimation runs and find that the surrogate accurately recovers the parameters of injected waveforms.

The tidal parameters of TRAPPIST-1b and c

Abstract The TRAPPIST-1 planetary system consists of seven planets within 0.05 au of each other, five of which are in a multiresonant chain. These resonances suggest the system formed via planet migration; subsequent tidal evolution has damped away most of the initial eccentricities. We used dynamical N-body simulations to estimate how long it takes for the multiresonant configuration that arises during planet formation to break. From there we use secular theory to pose limits on the tidal parameters of planets b and c. We calibrate our results against multilayered interior models constructed to fit the masses and radii of the planets, from which the tidal parameters are computed independently. The dynamical simulations show that the planets typically go unstable 30 Myr after their formation. Assuming synchronous rotation throughout, we compute $\frac{k_2}{Q} \gtrsim 2\times 10^{-4}$ for planet b and $\frac{k_2}{Q} \gtrsim 10^{-3}$ for planet c. Interior models yield (0.075–0.37) × 10−4 for TRAPPIST-1b and (0.4–2) × 10−4 for TRAPPIST-1c. The agreement between the dynamical and interior models is not too strong, but is still useful to constrain the dynamical history of the system. We suggest that this two-pronged approach could be of further use in other multiresonant systems if the planet’s orbital and interior parameters are sufficiently well known.