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DOE OSTI2025

VARI3D & PERSENT: Perturbation and Sensitivity Analysis

The nodal diffusion method is one of the most widely used approaches in modern reactor analysis. In the nodal diffusion method, a coarse multi-group set of “homogenized” parameters is constructed such that the complex geometry of a reactor core along with the energy dependence of neutron and gamma ray cross sections in a nuclear reactor are conserved in the simpler geometry. The homogenization is typically done on a fuel assembly level as is the case in the DIF3D code developed at Argonne National Laboratory. The nodal methodology is used primarily to predict fuel cycle behavior of nuclear systems of which there is a substantial amount of validation in the literature. Another use of the nodal method is to obtain reactivity coefficients and kinetics parameters for use in a safety analysis of a given nuclear reactor. While there are many ways to obtain reactivity worth and kinetics parameters, the work presented in this manuscript is unique as it provides the user with the ability to compute reactivity worths, kinetics parameters, and cross section sensitivities with a Cartesian and hexagonal geometry based transport code. This manuscript serves as a single manual for two separate codes: VARI3D and PERSENT. The VARI3D code (VARIational 3D) is based upon the classic finite difference diffusion theory solver available in DIF3D. The PERSENT code (PERturbation and SENitivity for Transport) is based upon the variational nodal method employed in DIF3D termed VARIANT. The VARIANT solver was added to DIF3D in 1995 and has seen continued development and use for the last 18 years. Because VARI3D primarily uses deprecated coding practices, rather than incorporating the perturbation and sensitivity treatments for transport within VARI3D, a new coding development was built using modern Fortran coding. The primary purpose of this manual is to describe the theory behind PERSENT (and by convenience, that of VARI3D) and discuss the input and output of PERSENT along with giving potential users an idea of how to use it. While this manuscript does describe the input and output of VARI3D, the PERSENT code is intended to be the replacement capability of VARI3D as PERSENT can generate nearly identical (if not superior) diffusion theory results. In this manuscript, the relevant aspects of generalized perturbation theory and exact perturbation theory that apply to both VARI3D and PERSENT are covered. The input and output of VARI3D is displayed by excerpting several of the example problems. Similarly, the input and output of PERSENT is displayed along with tips on how best to use the code. Note that the input and output of the inhomogeneous solver wrapped around DIF3D (DIF3D_IFS) is also discussed as it is needed to carry out some of the sensitivities in PERSENT such as reaction rate ratios. This manuscript describes several perturbation and sensitivity problems, and the results computed using PERSENT. From these sections, potential users should find that PERSENT provides not only the typical tables of numbers desired in perturbation and sensitivity analysis work, but also can visually plot the result for a more thorough understanding of the space and energy distribution (Section 5). Overall, PERSENT is observed to produce accurate reactivity worths and sensitivities for the displayed set of test problems and clearly demonstrates the need to have a transport-based sensitivity capability as evident from the thousands of percent errors observed in the 21-group hexagonal fast reactor problem (covered in Section 7). The uncertainty calculation capability is described in Section 3 and demonstrated in Section 7.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1999

Hydrodynamic Instability and Thermal Coupling in a Dynamic Model of Liquid-Propellant Combustion

For liquid-propellant combustion, the Landau/Levich hydrodynamic models have been combined and extended to account for a dynamic dependence of the burning rate on the local pressure and temperature fields. Analysis of these extended models is greatly facilitated by exploiting the realistic smallness of the gas-to-liquid density ratio rho. Neglecting thermal coupling effects, an asymptotic expression was then derived for the cellular stability boundary A(sub p)(k) where A(sub p) is the pressure sensitivity of the burning rate and k is the disturbance wavenumber. The results explicitly indicate the stabilizing effects of gravity on long-wave disturbances, and those of viscosity and surface tension on short-wave perturbations, and the instability associated with intermediate wavenumbers for critical negative values of A(sub p). In the limit of weak gravity, hydrodynamic instability in liquid-propellant combustion becomes a long-wave, instability phenomenon, whereas at normal gravity, this instability is first manifested through O(1) wavenumbers. In addition, surface tension and viscosity (both liquid and gas) each produce comparable effects in the large-wavenumber regime, thereby providing important modifications to the previous analyses in which one or more of these effects was neglected. For A(sub p)= O, the Landau/Levich results are recovered in appropriate limiting cases, although this typically corresponds to a hydrodynamically unstable parameter regime for p << 1. In addition to the classical cellular form of hydrodynamic stability, there exists a pulsating form corresponding to the loss of stability of steady, planar burning to time-dependent perturbations. This occurs for negative values of the parameter A(sub p), and is thus absent from the original Landau/Levich models. In the extended model, however, there exists a stable band of negative pressure sensitivities bounded above by the Landau type of instability, and below by this pulsating form of hydrodynamic instability. Indeed, nonsteady modes of combustion have been observed at low pressures in hydroxylammonium nitrate (HAN)-based liquid propellants, which often exhibit negative pressure sensitivities. While nonsteady combustion may correspond to secondary and higher-order bifurcations above the cellular boundary, it may also be a manifestation of this pulsating type of hydrodynamic instability. In the present work, a nonzero temperature sensitivity is incorporated into our previous asymptotic analyses. This entails a coupling of the energy equation to the previous purely hydrodynamic problem, and leads to a significant modification of the pulsating boundary such that, for sufficiently large values of the temperature-sensitivity parameter, liquid-propellant combustion can become intrinsically unstable to this alternative form of hydrodynamic instability. For simplicity, further attention is confined here to the inviscid version of the problem since, despite the fact that viscous and surface-tension effects are comparable, the qualitative nature of the cellular boundary remains preserved in the zero-viscosity limit, as does the existence of the pulsating boundary. The mathematical model adopts the classical assumption that there is no distributed reaction in either the liquid or gas phases, but now the reaction sheet, representing either a pyrolysis reaction or an exothermic decomposition at the liquid/gas interface, is assumed to depend on local conditions there.

Margolis, S. B.↗

NASA NTRS2004

Assessing the Usefulness of AIRS Radiance Observations in a 4D-Var Assimilation Scheme Using the Penn State/NCAR Mesoscale Model Version 5 (MM5) and a Stand Alone Radiative Transfer Algorithm (SARTA)

Short-term precipitation forecast skill remains poor despite further advances in numerical modeling and data analysis. Utilizing indirect observations, such as the radiance observations obtained from the Atmospheric Infrared Sounder (AIRS), in a 4D-Var assimilation scheme may lead to a rapid improvement in the skill of 12-h QPF. In order to investigate the usefulness of AIRS observations, the adjoint code of a radiative transfer model (SARTA) is developed and linked to MM5 4D-Var system. A statistical comparison of the AIRS radiances with those calculated using the linked MMS-SARTA model is carried out to assess errors in the radiative transfer model, MM5 and AIRS data, and to develop a quality control scheme. Outliers and any bias that may exist in radiative transfer model, MM5 and AIRS data are identified. An adjoint sensitivity study, utilizing the SARTA adjoint model, is then done to identify the relative sensitivities of each spectral channel to such parameters as atmospheric temperature, pressure, and mixing ratio. Another sensitivity study is done using the MM5 alone to determine the sensitivity of the 12-h QPF to model parameters such as atmospheric temperature, pressure, and mixing ratio under various weather regimes (i.e. summertime convection, winter snowfall). Results from these two types of adjoint sensitivity studies are used to determine which AIRS channels contribute most to 12-h QPF under differing weather regimes. This information is useful when conducting 4D-Var experiments using a subset of the 2378 AIRS spectral channels. Finally, 4D-Var experiments are done to analyze the impact of the AIRS observations on the quality of the 12-h QPF results. This project falls under the data assimilation general session topic as specified by the conference organizers.

Carrier, M.↗

NASA NTRS1982

Sensitivity analysis of helicopter IMC decelerating steep approach and landing performance to navigation system parameters

Results of a study to investigate, by means of a computer simulation, the performance sensitivity of helicopter IMC DSAL operations as a function of navigation system parameters are presented. A mathematical model representing generically a navigation system is formulated. The scenario simulated consists of a straight in helicopter approach to landing along a 6 deg glideslope. The deceleration magnitude chosen is 03g. The navigation model parameters are varied and the statistics of the total system errors (TSE) computed. These statistics are used to determine the critical navigation system parameters that affect the performance of the closed-loop navigation, guidance and control system of a UH-1H helicopter.

Karmali, M. S.↗

NASA NTRS1990

Derivatives of buckling loads and vibration frequencies with respect to stiffness and initial strain parameters

A uniform variational approach to sensitivity analysis of vibration frequencies and bifurcation loads of nonlinear structures is developed. Two methods of calculating the sensitivities of bifurcation buckling loads and vibration frequencies of nonlinear structures, with respect to stiffness and initial strain parameters, are presented. A direct method requires calculation of derivatives of the prebuckling state with respect to these parameters. An adjoint method bypasses the need for these derivatives by using instead the strain field associated with the second-order postbuckling state. An operator notation is used and the derivation is based on the principle of virtual work. The derivative computations are easily implemented in structural analysis programs. This is demonstrated by examples using a general purpose, finite element program and a shell-of-revolution program.

Haftka, Raphael T.↗

DOE OSTI2024

VARI3D & PERSENT: Perturbation and Sensitivity Analysis (Revision 5)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1986

Design concepts/parameters assessment and sensitivity analyses of select composite structural components

Formal approaches are summarized to evaluate design concepts and perform sensitivity analyses on design parameters of composite structural components for vehicles. The formal approaches include structural analyses coupled with composite micromechanics to assess the structural response of beams made from various intraply hybrids, finite element analysis in conjunction with composite mechanics to assess the structural response of panels made from strip hybrids, and sensitivity analysis through optimization to assess the effects of various design parameters on the optimum design of a panel made from angleplied composite laminates. Results obtained from these approaches are presented in graphical and tabular form to illustrate parametric studies and acceptable ranges of various design parameters.

Chamis, C. C.↗

NASA NTRS1980

Design and breadboard evaluation of the SPS reference phase control system concept

The total breadboard system includes one pilot transmitter, one pilot receiver, nine phase distribution units, and two power transponders. With this complement of equipment, segments of a typical phase distribution system can be assembled to facilitate the evaluation of significant system parameters. The achievable accuracy of a large phase distribution system, the sensitivity of the system to parameter variations, and the limitations of commercially available components in such applications were determined.

Hopkins, P. M.↗

NASA NTRS2017

Sensitivity of Impact Risk to Uncertainty in Asteroid Properties and Entry Parameters

A central challenge in evaluating the threat posed by asteroids striking Earth is the large amount of uncertainty in potential asteroid properties and entry parameters, which can vary the resulting ground damage and affected population by orders of magnitude. We are using our Probabilistic Asteroid Impact Risk (PAIR) model to investigate the sensitivity of asteroid impact damage to these uncertainties. To assess the risk sensitivity, we alternately fix or vary the different input parameters and compare the damage distributions produced. In this study, we consider local ground damage from blast waves or thermal radiation for impactors 50-500m in diameter. The ongoing goal of this work is to help guide future efforts in asteroid characterization and model refinement by determining which properties most significantly affect the potential risk.

Wheeler, Lorien↗

NASA NTRS1997

Validation of an Accurate Three-Dimensional Helical Slow-Wave Circuit Model

The helical slow-wave circuit embodies a helical coil of rectangular tape supported in a metal barrel by dielectric support rods. Although the helix slow-wave circuit remains the mainstay of the traveling-wave tube (TWT) industry because of its exceptionally wide bandwidth, a full helical circuit, without significant dimensional approximations, has not been successfully modeled until now. Numerous attempts have been made to analyze the helical slow-wave circuit so that the performance could be accurately predicted without actually building it, but because of its complex geometry, many geometrical approximations became necessary rendering the previous models inaccurate. In the course of this research it has been demonstrated that using the simulation code, MAFIA, the helical structure can be modeled with actual tape width and thickness, dielectric support rod geometry and materials. To demonstrate the accuracy of the MAFIA model, the cold-test parameters including dispersion, on-axis interaction impedance and attenuation have been calculated for several helical TWT slow-wave circuits with a variety of support rod geometries including rectangular and T-shaped rods, as well as various support rod materials including isotropic, anisotropic and partially metal coated dielectrics. Compared with experimentally measured results, the agreement is excellent. With the accuracy of the MAFIA helical model validated, the code was used to investigate several conventional geometric approximations in an attempt to obtain the most computationally efficient model. Several simplifications were made to a standard model including replacing the helical tape with filaments, and replacing rectangular support rods with shapes conforming to the cylindrical coordinate system with effective permittivity. The approximate models are compared with the standard model in terms of cold-test characteristics and computational time. The model was also used to determine the sensitivity of various circuit parameters including typical manufacturing dimensional tolerances and support rod permittivity. By varying the circuit parameters of an accurate model using MAFIA, these sensitivities can be computed for manufacturing concerns, and design optimization previous to fabrication, thus eliminating the need for costly experimental iterations. Several variations were made to a standard helical circuit using MAFIA to investigate the effect that variations on helical tape and support rod width, metallized loading height and support rod permittivity, have on TWT cold-test characteristics.

Kory, Carol L.↗

NASA NTRS2004

Employing Sensitivity Derivatives to Estimate Uncertainty Propagation in CFD

Two methods that exploit the availability of sensitivity derivatives are successfully employed to predict uncertainty propagation through Computational Fluid Dynamics (CFD) code for an inviscid airfoil problem. An approximate statistical second-moment method and a Sensitivity Derivative Enhanced Monte Carlo (SDEMC) method are successfully demonstrated on a two-dimensional problem. First- and second-order sensitivity derivatives of code output with respect to code input are obtained through an efficient incremental iterative approach. Given uncertainties in statistically independent, random, normally distributed flow parameters (input variables); these sensitivity derivatives enable one to formulate first- and second-order Taylor Series approximations for the mean and variance of CFD output quantities. Additionally, incorporation of the first-order sensitivity derivatives into the data reduction phase of a conventional Monte Carlo (MC) simulation allows for improved accuracy in determining the first moment of the CFD output. Both methods are compared to results generated using a conventional MC method. The methods that exploit the availability of sensitivity derivatives are found to be valid when considering small deviations from input mean values.

Putko, Michele M.↗

NASA NTRS2018

The Effects of Uncertainty in Initial CME Input Parameters on Deflection, Rotation, Bz, and Arrival Time Predictions

Understanding the effects of coronal mass ejections (CMEs) requires knowing if and when they will impact and their properties upon impact. Of particular importance is the strength of a CME's southward magnetic field component (Bz ). Kay et al. (2013, https://doi:10.1088/0004-637X/775/1/5, 2015, https://doi:10.1088/948 0004-637X/805/2/168) have shown that the simplified analytic model Forecasting a CME's Altered Trajectory (ForeCAT) can reproduce the deflection and rotation of CMEs. Kay, Gopalswamy, Reinard, and Opher (2017, https://doi.org/10.3847/1538-4357/835/2/117) introduced ForeCAT In situ Data Observer, which uses ForeCAT results to simulate magnetic field profiles. ForeCAT In situ Data Observer reproduces the in situ observations on roughly hourly time scales, suggesting that these models could be extremely useful for predictions of Bz . However, as with all models, both models are sensitive to their input parameters, which may not be precisely known for predictions. We explore this sensitivity using ensembles having small changes in the initial latitude, longitude, and orientation of the erupting CME. We explore the effects of different background magnetic field models and find that the changes in deflection and rotation resulting from the uncertainty in the initial parameters tend to exceed the changes from dierent magnetic backgrounds. The range in the in situ proles tends to scale with the range in the deflection and rotation. We also consider a simple arrival time model using ForeCAT results and nd an average absolute error of only 3 hr. We show that an uncertainty in the CME position of 8.1 ± 6.9 leads to variations of 6 hr in the arrival time. This measure depends strongly on the location of impact within the CME with the arrival time changing less for impacts near the nose.

Kay, Christina D.↗

NASA NTRS1986

Oceanic cloud feedbacks on earth radiation budget parameters

Oceanic cloud variability and sensitivity to sea surface temperature (SST) were examined using radiometer data gathered with instrumentation on the Nimbus-7 satellite. The study area was a region of the Pacific Ocean north of Hawaii. SST and albedo data for the period 1978-1983 were compared and similar trends were found for minimal values of the albedo and SST.

Chertock, B.↗

NASA NTRS2017

Vertical Structure and Optical Properties of Titans Aerosols from Radiance Measurements Made Inside and Outside the Atmosphere

Prompted by the detection of stratospheric cloud layers by Cassini's Composite Infrared Spectrometer (CIRS; see Anderson, C.M., Samuelson, R.E. [2011]. Icarus 212, 762-778), we have re-examined the observations made by the Descent Imager/Spectral Radiometer (DISR) in the atmosphere of Titan together with two constraints from measurements made outside the atmosphere. No evidence of thin layers (<1 km) in the DISR image data sets is seen beyond the three previously reported layers at 21 km, 11 km, and 7 km by Karkoschka and Tomasko (Karkoschka, E., Tomasko, M.G. [2009]. Icarus 199, 442-448). On the other hand, there is evidence of a thicker layer centered at about 55 km. A rise in radiance gradients in the Downward-Looking Visible Spectrometer (DLVS) data below 55 km indicates an increase in the volume extinction coefficient near this altitude. To fit the geometric albedo measured from outside the atmosphere the decrease in the single scattering albedo of Titan's aerosols at high altitudes, noted in earlier studies of DISR data, must continue to much higher altitudes. The altitude of Titan's limb as a function of wavelength requires that the scale height of the aerosols decrease with altitude from the 65 km value seen in the DISR observations below 140 km to the 45 km value at higher altitudes. We compared the variation of radiance with nadir angle observed in the DISR images to improve our aerosol model. Our new aerosol model fits the altitude and wavelength variations of the observations at small and intermediate nadir angles but not for large nadir angles, indicating an effect that is not reproduced by our radiative transfer model. The volume extinction profiles are modeled by continuous functions except near the enhancement level near 55 km altitude. The wavelength dependence of the extinction optical depth is similar to earlier results at wavelengths from 500 to 700 nm, but is smaller at shorter wavelengths and larger toward longer wavelengths. A Hapke-like model is used for the ground reflectivity, and the variation of the Hapke single scattering albedo with wavelength is given. Fits to the visible spectrometers looking upward and downward are achieved except in the methane bands longward of 720 nm. This is possibly due to uncertainties in extrapolation of laboratory measurements from 1 km-am paths to much longer paths at lower pressures. It could also be due to changes in the single scattering phase functions at low altitudes, which strongly affect the path length through methane that the photons travel. We demonstrate the effects on the model fits by varying each model parameter individually in order to illustrate the sensitivity of our determination of each model parameter.

Titan↗

NASA NTRS2022

Earth System Model Parameter Adjustment Using a Green's Functions Approach

We demonstrate the practicality and effectiveness of using a Green's functions estimation approach for adjusting uncertain parameters in an Earth system model (ESM). This estimation approach has previously been applied to an intermediate-complexity climate model and to individual ESM components, e.g., ocean, sea ice, or carbon cycle components. Here, the Green's functions approach is applied to a state-of-the-art ESM that comprises a global atmosphere/land configuration of the Goddard Earth Observing System (GEOS) coupled to an ocean and sea ice configuration of the Massachusetts Institute of Technology general circulation model (MITgcm). Horizontal grid spacing is approximately 110 km for GEOS and 37–110 km for MITgcm. In addition to the reference GEOS-MITgcm simulation, we carried out a series of model sensitivity experiments, in which 20 uncertain parameters are perturbed. These “control” parameters can be used to adjust sea ice, microphysics, turbulence, radiation, and surface schemes in the coupled simulation. We defined eight observational targets: sea ice fraction, net surface shortwave radiation, downward longwave radiation, near-surface temperature, sea surface temperature, sea surface salinity, and ocean temperature and salinity at 300 m. We applied the Green's functions approach to optimize the values of the 20 control parameters so as to minimize a weighted least-squares distance between the model and the eight observational targets. The new experiment with the optimized parameters resulted in a total cost reduction of 9 % relative to a simulation that had already been adjusted using other methods. The optimized experiment attained a balanced cost reduction over most of the observational targets. We also report on results from a set of sensitivity experiments that are not used in the final optimized simulation but helped explore options and guided the optimization process. These experiments include an assessment of sensitivity to the number of control parameters and to the selection of observational targets and weights in the cost function. Based on these sensitivity experiments, we selected a specific definition for the cost function. The sensitivity experiments also revealed a decreasing overall cost as the number of control variables was increased. In summary, we recommend using the Green's functions estimation approach as an additional fine-tuning step in the model development process. The method is not a replacement for modelers' experience in choosing and adjusting sensitive model parameters. Instead, it is an additional practical and effective tool for carrying out final adjustments of uncertain ESM parameters.

Green's Function↗

NASA NTRS1993

Space transportation vehicle design evaluation using saturated designs

An important objective in the preliminary design and evaluation of space transportation vehicles is to find the best values of design variables that optimize the performance characteristic (e.g. dry weight). For a given configuration, the vehicle performance can be determined by the use of complex sizing and performance evaluation computer programs. These complex computer programs utilize iterative algorithms and they are generally too expensive and/or difficult to use directly in multidisciplinary design optimization. An alternative is to use response surface methodology (RSM) and obtain quadratic polynomial approximations to the functional relationships between performance characteristics and design variables. In RSM, these approximation models are then used to determine optimum design parameter values and for rapid sensitivity studies. Constructing a second-order model requires that 'n' design parameters be studied at least at 3 levels (values) so that the coefficients in the model can be estimated. There, 3(n) factorial experiments (point designs or observations) may be necessary. For small values of 'n' such as two or three, this design works well. However, when a large number of design parameters are under study, the number of design points required for a full-factorial design may become excessive. Fortunately, these quadratic polynomial approximations can be obtained by selecting an efficient design matrix using central composite designs (CCD) from design of experiments theory. Each unique point design from the CCD matrix is then conducted using computerized analysis tools (e.g. POST, CONSIZ, etc.). In the next step, least squares regression analysis is used to calculate the quadratic polynomial coefficients from the data. However, in some multidisciplinary applications involving a large number of design variables and several disciplines, the computerized performance synthesis programs may get too time consuming and expensive to run even with the use of efficient central composite designs. In such cases, it may be preferable to keep the number of design points to an absolute minimum and trade some model accuracy with cost. For this purpose, another class of experimental designs, called saturated D-optimal designs may be utilized for generating a matrix of vehicle designs. A design is called saturated when the number of design points is exactly equal to the number of terms in the model to be fitted. As a result, saturated designs require the absolute minimum number of design points ((n+1)(n+2)/s) to estimate the quadratic polynomial model coefficients. Saturated designs can be generated using the D-optimality criterion. A good saturated design should give rise to least squares estimates with minimum generalized variance.

Unal, Resit↗

NASA NTRS1990

VIP: A knowledge-based design aid for the engineering of space systems

The Vehicles Implementation Project (VIP), a knowledge-based design aid for the engineering of space systems is described. VIP combines qualitative knowledge in the form of rules, quantitative knowledge in the form of equations, and other mathematical modeling tools. The system allows users rapidly to develop and experiment with models of spacecraft system designs. As information becomes available to the system, appropriate equations are solved symbolically and the results are displayed. Users may browse through the system, observing dependencies and the effects of altering specific parameters. The system can also suggest approaches to the derivation of specific parameter values. In addition to providing a tool for the development of specific designs, VIP aims at increasing the user's understanding of the design process. Users may rapidly examine the sensitivity of a given parameter to others in the system and perform tradeoffs or optimizations of specific parameters. A second major goal of VIP is to integrate the existing corporate knowledge base of models and rules into a central, symbolic form.

Lewis, Steven M.↗

NASA NTRS2021

A Uniform Search for Nearby Planetary Companions to Hot Jupiters in TESS Data Reveals Hot Jupiters Are Still Lonely

We present the results of a uniform search for additional planets around all stars with confirmed hot Jupiters observed by the Transiting Exoplanet Survey Satellite (TESS) in its Cycle 1 survey of the southern ecliptic hemisphere. Our search comprises 184 total planetary systems with confirmed hot Jupiters with Rp > 8 R⊕ and orbital period <10 days. The Transit Least Squares algorithm was utilized to search for periodic signals that may have been missed by other planet search pipelines. While we recovered 169 of these confirmed hot Jupiters, our search yielded no new statistically validated planetary candidates in the parameter space searched (P < 14 days). A lack of planet candidates nearby hot Jupiters in the TESS data supports results from previous transit searches of each individual system, now down to the photometric precision of TESS. This is consistent with expectations from a high-eccentricity migration formation scenario, but additional formation indicators are needed for definitive confirmation. We injected transit signals into the light curves of the hot Jupiter sample to probe the pipeline's sensitivity to the target parameter space, finding a dependence proportional to ${R}_{p}^{2.32}{P}^{-0.88}$ for planets within 0.3 ≤ Rp ≤ 4 R⊕ and 1 ≤ P ≤ 14 days. A statistical analysis accounting for this sensitivity provides a median and 90% confidence interval of ${7.3}_{-7.3}^{+15.2} \% $ for the rate of hot Jupiters with nearby companions in this target parameter space. This study demonstrates how TESS uniquely enables comprehensive searches for nearby planetary companions to nearly all the known hot Jupiters.

Benjamin J Hord↗