arXiv subjects

B. Scott Gaudi

Publications and source records attributed to B. Scott Gaudi.

At least 37 records · Page 2Linked to original sources

arXiv2024

Exploring Systematic Errors in the Inferred Parameters of the Transiting Planet KELT-15b and its Host Star

Transiting planet systems offer a unique opportunity to measure the masses and radii of many planets and their host stars. Yet, relative photometry and radial velocity measurements alone only constrain the density of the host star. In remedy, the community uses theoretical and semi-empirical methods to break this one-parameter degeneracy and measure the mass and radius of the host star and its planet(s). We investigate the differences in the inferred system parameters due to modeling a host star with the Torres mass-radius relations, YY evolutionary tracks, MIST evolutionary tracks, and a stellar radius estimate from the spectral energy distribution (SED). We consider the effects of different priors on the stellar effective temperature, limb darkening, and eccentricity of the planet. Using the publicly available software package EXOFASTv2, we globally model TESS photometry and radial velocity observations of KELT-15, which hosts a fairly representative hot Jupiter. In total, we explore the impact of 28 different choices of priors on the inferred parameters of KELT-15b. We find broad agreement in the inferred system parameters across methodologies at the level of ~1.1 sigma between the MIST and SED constraints. This gives some confidence that systematic errors are not ubiquitous in transiting planets systems. We also find a ~2 sigma difference in the stellar radius estimated by the MIST models when we adopt differing literature spectroscopic effective temperature estimates. Similar studies of a large number of systems are needed to definitely assess systematic uncertainties the exoplanet population as a whole.

astro-ph.EP↗

arXiv2024

OGLE-2015-BLG-0845L: A low-mass M dwarf from the microlensing parallax and xallarap effects

We present the analysis of the microlensing event OGLE-2015-BLG-0845, which was affected by both the microlensing parallax and xallarap effects. The former was detected via the simultaneous observations from the ground and Spitzer, and the latter was caused by the orbital motion of the source star in a relatively close binary. The combination of these two effects led to a mass measurement of the lens object, revealing a low-mass ($0.14 \pm 0.05 M_{\odot}$) M-dwarf at the bulge distance ($7.6 \pm 1.0$ kpc). The source binary consists of a late F-type subgiant and a K-type dwarf of $\sim1.2 M_{\odot}$ and $\sim 0.9 M_{\odot}$, respectively, and the orbital period is $70 \pm 10$ days. OGLE-2015-BLG-0845 is the first single-lens event in which the lens mass is measured via the binarity of the source. Given the abundance of binary systems as potential microlensing sources, the xallarap effect may not be a rare phenomenon. Our work thus highlights the application of the xallarap effect in the mass determination of microlenses, and the same method can be used to identify isolated dark lenses.

astro-ph.SR↗

arXiv2024

The EBLM project X. Benchmark masses, radii and temperatures for two fully convective M-dwarfs using K2

M-dwarfs are the most abundant stars in the galaxy and popular targets for exoplanet searches. However, their intrinsic faintness and complex spectra inhibit precise characterisation. We only know of dozens of M-dwarfs with fundamental parameters of mass, radius and effective temperature characterised to better than a few per cent. Eclipsing binaries remain the most robust means of stellar characterisation. Here we present two targets from the Eclipsing Binary Low Mass (EBLM) survey that were observed with K2: EBLM J0055-00 and EBLM J2217-04. Combined with HARPS and CORALIE spectroscopy, we measure M-dwarf masses with precisions better than 5%, radii better than 3% and effective temperatures on order 1%. However, our fits require invoking a model to derive parameters for the primary star. By investigating three popular models, we determine that the model uncertainty is of similar magnitude to the statistical uncertainty in the model fits. Therefore, whilst these can be considered benchmark M-dwarfs, we caution the community to consider model uncertainty when pushing the limits of precise stellar characterisation.

astro-ph.SR↗

arXiv2023

The PEPSI Exoplanet Transit Survey (PETS) IV: Assessing the atmospheric chemistry of KELT-20b

Most ultra hot Jupiters (UHJs) show evidence of temperature inversions, in which temperature increases with altitude over a range of pressures. Temperature inversions can occur when there is a species that absorbs the stellar irradiation at a relatively high level of the atmospheres. However, the species responsible for this absorption remains unidentified. In particular, the UHJ KELT-20b is known to have a temperature inversion. Using high resolution emission spectroscopy from LBT/PEPSI we investigate the atomic and molecular opacity sources that may cause the inversion in KELT-20b, as well as explore its atmospheric chemistry. We confirm the presence of Fe I with a significance of 17$σ$. We also report a tentative $4.3σ$ detection of Ni I. A nominally $4.5σ$ detection of Mg I emission in the PEPSI blue arm is likely in fact due to aliasing between the Mg I cross-correlation template and the Fe I lines present in the spectrum. We cannot reproduce a recent detection of Cr I, while we do not have the wavelength coverage to robustly test past detections of Fe II and Si I. Together with non-detections of molecular species like TiO, this suggests that Fe I is likely to be the dominant optical opacity source in the dayside atmosphere of KELT-20b and may be responsible for the temperature inversion. We explore ways to reconcile the differences between our results and those in literature and point to future paths to understand atmospheric variability.

astro-ph.EP↗

arXiv2023

A Comparison of the Composition of Planets in Single- and Multi-Planet Systems Orbiting M dwarfs

We investigate and compare the composition of M-dwarf planets in systems with only one known planet (``singles") to those residing in multi-planet systems (``multis") and the fundamental properties of their host stars. We restrict our analysis to planets with directly measured masses and radii, which comprise a total of 70 planets: 30 singles and 40 multis in 19 systems. We compare the bulk densities for the full sample, which includes planets ranging in size from $0.52 R_{\oplus}$ to $12.8R_\oplus$, and find that single planets have significantly lower densities on average than multis, which we cannot attribute to selection biases. We compare the bulk densities normalized by an Earth model for planets with $R_{p} < 6R_{\oplus}$, and find that multis are also denser with 99\% confidence. We calculate and compare the core/water mass fractions (CMF/WMF) of low-mass planets ($M_p <10 M_{\oplus}$), and find that the likely rocky multis (with $R_p <1.6 R_{\oplus}$) have lower CMFs than singles. We also compare the [Fe/H] metallicity and rotation period of all single versus multi-planet host stars with such measurements in the literature and find that multi-planet hosts are significantly more metal-poor than those hosting a single planet. Moreover, we find that host star metallicity decreases with increasing planet multiplicity. In contrast, we find only a modest difference in the rotation period. The significant differences in planetary composition and metallicity of the host stars point to different physical processes governing the formation of single- and multi-planet systems in M dwarfs.

astro-ph.EP↗

arXiv2023

Roman CCS White Paper: Adding Fields Hosting Globular Clusters To The Galactic Bulge Time Domain Survey

Despite multiple previous searches, no transiting planets have yet been identified within a globular cluster. This is believed to be due to a combination of factors: the low metallicities of most globular clusters suggests that there is significantly less planet-forming material per star in most globular clusters relative to the solar neighborhood, the high likelihood of dynamical interactions can also disrupt planetary orbits, and the data available for globular clusters is limited. However, transiting planets have been identified in open clusters, indicating that there may be planets in more massive clusters that have simply gone undetected, or that more massive clusters inhibit planet formation. Less than two degrees away from the nominal Galactic Bulge Time Domain Survey footprint, two globular clusters, NGC 6522 and NGC 6528, can be simultaneously observed by the Roman telescope during the Galactic Bulge Time Domain Survey. These clusters are comparable in mass (1-2 x 10$^5$ solar masses) and age (12 Gyr), but feature drastically different average metallicities: NGC 6522 has an average [Fe/H] $\sim$ -1.3, while NGC 6528 has an average [Fe/H] $\sim$ -0.1. If no transiting planets are detected in one season of time domain observations of these clusters, this would indicate a difference in planet occurrence among field stars and globular clusters at >3-$σ$ significance even after accounting for metallicity, which could be enhanced to >5-$σ$ significance with similar observations of another nearby field hosting a metal-rich globular cluster. Additionally, time domain observations of NGC 6522 and NGC 6528 will detect variable stars in both clusters, testing the connection between stellar variability and binary fraction to metallicity and cluster environment, as well as testing the dependence of exoplanet yields on stellar density and distance from the Galactic midplane.

astro-ph.IM↗

arXiv2023

K2 Optical Emission from OJ 287 and Other Gamma-Ray Blazars on Hours-to-Weeks Timescales from 2014-2018

We present second observations by K2 of OJ~287 and 7 other $γ$-ray AGNs obtained in 2017-2018, second and third observations of the lobe-dominated, steep spectrum quasar 3C~207, and observations of 9 additional blazars not previously observed with K2. The AGN were observed simultaneously with K2 and the Fermi Large Area Telescope for 51-81 days. Our full sample, observed in 2014-2018, contained 16 BL Lac objects (BL Lacs), 9 Flat Spectrum Radio Quasars (FSRQs), and 4 other $γ$-ray AGNs. Twelve BL Lacs and 7 FSRQs exhibited fast, jagged light curves while 4 BL Lacs and 2 FSRQs had slow, smooth light curves. Some objects changed their red-noise character significantly between repeated K2 observations. The optical characteristics of OJ~287 derived from the short-cadence K2 light curves changed between observations made before and after the predicted passage of the suspected secondary supermassive black hole through the accretion disk of the primary supermassive black hole. The average slopes of the periodogram power spectral densities of the BL Lacs' and FSRQs' light curves differed significantly, by $\approx 12$\%, with the BL Lac slopes being steeper, and a KS test with a $p$-value of 0.039 indicates that these samples probably come from different populations; however, this result is not as strongly supported by PSRESP analyses. Differences in the origin of the jets from the ergosphere or accretion disk in these two classes could produce such a disparity, as could different sizes or locations of emission regions within the jets.

astro-ph.HE↗

arXiv2023

Another Shipment of Six Short-Period Giant Planets from TESS

We present the discovery and characterization of six short-period, transiting giant planets from NASA's Transiting Exoplanet Survey Satellite (TESS) -- TOI-1811 (TIC 376524552), TOI-2025 (TIC 394050135), TOI-2145 (TIC 88992642), TOI-2152 (TIC 395393265), TOI-2154 (TIC 428787891), & TOI-2497 (TIC 97568467). All six planets orbit bright host stars (8.9 <G< 11.8, 7.7 <K< 10.1). Using a combination of time-series photometric and spectroscopic follow-up observations from the TESS Follow-up Observing Program (TFOP) Working Group, we have determined that the planets are Jovian-sized (R$_{P}$ = 1.00-1.45 R$_{J}$), have masses ranging from 0.92 to 5.35 M$_{J}$, and orbit F, G, and K stars (4753 $<$ T$_{eff}$ $<$ 7360 K). We detect a significant orbital eccentricity for the three longest-period systems in our sample: TOI-2025 b (P = 8.872 days, $e$ = $0.220\pm0.053$), TOI-2145 b (P = 10.261 days, $e$ = $0.182^{+0.039}_{-0.049}$), and TOI-2497 b (P = 10.656 days, $e$ = $0.196^{+0.059}_{-0.053}$). TOI-2145 b and TOI-2497 b both orbit subgiant host stars (3.8 $<$ $\log$ g $<$4.0), but these planets show no sign of inflation despite very high levels of irradiation. The lack of inflation may be explained by the high mass of the planets; $5.35^{+0.32}_{-0.35}$ M$_{\rm J}$ (TOI-2145 b) and $5.21\pm0.52$ M$_{\rm J}$ (TOI-2497 b). These six new discoveries contribute to the larger community effort to use {\it TESS} to create a magnitude-complete, self-consistent sample of giant planets with well-determined parameters for future detailed studies.

astro-ph.EP↗

arXiv2023

Exoplanet Nodal Precession Induced by Rapidly Rotating Stars: Impacts on Transit Probabilities and Biases

For the majority of short period exoplanets transiting massive stars with radiative envelopes, the spin angular momentum of the host star is greater than the planetary orbital angular momentum. In this case, the orbits of the planets will undergo nodal precession, which can significantly impact the probability that the planets transit their parent star. In particular, for some combinations of the spin-orbit angle $ψ$ and the inclination of the stellar spin $i_*$, all such planets will eventually transit at some point over the duration of their precession period. Thus, as the time over which the sky has been monitored for transiting planets increases, the frequency of planets with detectable transits will increase, potentially leading to biased estimates of exoplanet occurrence rates, especially orbiting more massive stars. Furthermore, due to the dependence of the precession period on orbital parameters such as spin-orbit misalignment, the observed distributions of such parameters may also be biased. We derive the transit probability of a given exoplanet in the presence of nodal precession induced by a rapidly spinning host star. We find that the effect of nodal precession has already started to become relevant for some short-period planets, i.e., Hot Jupiters, orbiting massive stars, by increasing transit probabilities by of order a few percent for such systems within the original $Kepler$ field. We additionally derive simple expressions to describe the time evolution of the impact parameter $b$ for applicable systems, which should aid in future investigations of exoplanet nodal precession and spin-orbit alignment.

astro-ph.EP↗

arXiv2023

OGLE-2017-BLG-1038: A Possible Brown-dwarf Binary Revealed by Spitzer Microlensing Parallax

We report the analysis of microlensing event OGLE-2017-BLG-1038, observed by the Optical Gravitational Lensing Experiment, Korean Microlensing Telescope Network, and Spitzer telescopes. The event is caused by a giant source star in the Galactic Bulge passing over a large resonant binary lens caustic. The availability of space-based data allows the full set of physical parameters to be calculated. However, there exists an eightfold degeneracy in the parallax measurement. The four best solutions correspond to very-low-mass binaries near ($M_1 = 170^{+40}_{-50} M_J$ and $M_2 = 110^{+20}_{-30} M_J$), or well below ($M_1 = 22.5^{+0.7}_{-0.4} M_J$ and $M_2 = 13.3^{+0.4}_{-0.3} M_J$) the boundary between stars and brown dwarfs. A conventional analysis, with scaled uncertainties for Spitzer data, implies a very-low-mass brown dwarf binary lens at a distance of 2 kpc. Compensating for systematic Spitzer errors using a Gaussian process model suggests that a higher mass M-dwarf binary at 6 kpc is equally likely. A Bayesian comparison based on a galactic model favors the larger-mass solutions. We demonstrate how this degeneracy can be resolved within the next ten years through infrared adaptive-optics imaging with a 40 m class telescope.

astro-ph.SR↗

arXiv2023

The PEPSI Exoplanet Transit Survey (PETS). II. A Deep Search for Thermal Inversion Agents in KELT-20 b/MASCARA-2 b with Emission and Transmission Spectroscopy

Recent observations have shown that the atmospheres of ultra hot Jupiters (UHJs) commonly possess temperature inversions, where the temperature increases with increasing altitude. Nonetheless, which opacity sources are responsible for the presence of these inversions remains largely observationally unconstrained. We used LBT/PEPSI to observe the atmosphere of the UHJ KELT-20 b in both transmission and emission in order to search for molecular agents which could be responsible for the temperature inversion. We validate our methodology by confirming previous detections of Fe I in emission at $16.9σ$. Our search for the inversion agents TiO, VO, FeH, and CaH results in non-detections. Using injection-recovery testing we set $4σ$ upper limits upon the volume mixing ratios for these constituents as low as $\sim1\times10^{-9}$ for TiO. For TiO, VO, and CaH, our limits are much lower than expectations from an equilibrium chemical model, while we cannot set constraining limits on FeH with our data. We thus rule out TiO and CaH as the source of the temperature inversion in KELT-20 b, and VO only if the line lists are sufficiently accurate.

astro-ph.EP↗

arXiv2022

Is LTT 1445 Ab a Hycean World or a cold Haber World? Exploring the Potential of Twinkle to Unveil Its Nature

We explore the prospects for Twinkle to determine the atmospheric composition of the nearby terrestrial-like planet LTT 1445 Ab, including the possibility of detecting the potential biosignature ammonia (NH$_{3}$). At a distance of 6.9 pc, this system is the second closest known transiting system and will be observed through transmission spectroscopy with the upcoming Twinkle mission. Twinkle is equipped with a 0.45 m telescope, covers a spectral wavelength range of 0.5 - 4.5 $μ$m simultaneously with a resolving power between 50 - 70, and is designed to study exoplanets, bright stars, and solar system objects. We investigate the mission's potential to study LTT 1445 Ab and find that Twinkle data can distinguish between a cold Haber World (N$_2$-H$_2$-dominated atmosphere) and a Hycean World with a H$_2$O-H$_2$-dominated atmosphere, with a $χ_ν^{2}$ = 3.01. Interior composition analysis favors a Haber World scenario for LTT 1445 Ab, which suggests that the planet probably lacks a substantial water layer. We use petitRADTRANS and a Twinkle simulator to simulate transmission spectra for the more likely scenario of a cold Haber World for which NH$_{3}$ is considered to be a biosignature. We study the detectability under different scenarios: varying hydrogen fraction, concentration of ammonia, and cloud coverage. We find that ammonia can be detected at a $\sim$ 3$σ$ level for optimal (non-cloudy) conditions with 25 transits and a volume mixing ration of 4.0 ppm of NH$_{3}$. We provide examples of retrieval analysis to constrain potential NH$_{3}$ and H$_{2}$O in the atmosphere. Our study illustrates the potential of Twinkle to characterize atmospheres of potentially habitable exoplanets.

astro-ph.EP↗

arXiv2022

A Mathematical Treatment of the Offset Microlensing Degeneracy

The offset microlensing degeneracy, recently proposed by Zhang et al. (2022), has been shown to generalize the close-wide and inner-outer caustic degeneracies into a unified regime of magnification degeneracy in the interpretation of 2-body planetary microlensing observations. While the inner-outer degeneracy expects the source trajectory to pass equidistant to the planetary caustics of the degenerate lens configurations, the offset degeneracy states that the same mathematical expression applies to any combination of the close, wide, and resonant caustic topologies, where the projected star-planet separations differ by an offset ($s_{\rm A}\neq s_{\rm B}$) that depends on where the source trajectory crosses the star-planet axis. An important implication is that the $s_{\rm A}=1/s_{\rm B}$ solution of the close-wide degeneracy never strictly manifests in observations except when the source crosses a singular point near the primary. Nevertheless, the offset degeneracy was proposed upon numerical calculations, and no theoretical justification was given. Here, we provide a theoretical treatment of the offset degeneracy, which demonstrates its nature as a mathematical degeneracy. From first principles, we show that the offset degeneracy formalism is exact to zeroth-order in the mass ratio ($q$) for two cases: when the source crosses the lens-axis inside of caustics, and for $(s_{\rm A}-s_{\rm B})^6\ll1$ when crossing outside of caustics. The extent to which the offset degeneracy persists in oblique source trajectories is explored numerically. Lastly, it is shown that the superposition principle allows for a straightforward generalization to $N$-body microlenses with $N-1$ planetary lens components ($q\ll1$), which results in a $2^{N-1}$-fold degeneracy.

astro-ph.EP↗

arXiv2022

Revised Temperatures For Two Benchmark M-dwarfs -- Outliers No More

Well-characterised M-dwarfs are rare, particularly with respect to effective temperature. In this letter we re-analyse two benchmark M-dwarfs in eclipsing binaries from Kepler/K2: KIC 1571511AB and HD 24465AB. Both have temperatures reported to be hotter or colder by approximately 1000 K in comparison with both models and the majority of the literature. By modelling the secondary eclipses with both the original data and new data from TESS we derive significantly different temperatures which are not outliers. Removing this discrepancy allows these M-dwarfs to be truly benchmarks. Our work also provides relief to stellar modellers. We encourage more measurements of M-dwarf effective temperatures with robust methods.

astro-ph.SR↗

arXiv2022

A Reanalysis of the Composition of K2-106b: an Ultra-short Period Super-Mercury Candidate

We present a reanalysis of the K2-106 transiting planetary system, with a focus on the composition of K2-106b, an ultra-short period, super-Mercury candidate. We globally model existing photometric and radial velocity data and derive a planetary mass and radius for K2-106b of $M_{p} = 8.53\pm1.02~M_{\oplus}$ and $R_{p} = 1.71^{+0.069}_{-0.057}~R_{\oplus}$, which leads to a density of $ρ_{p} = 9.4^{+1.6}_{-1.5}$ $\rm g~cm^{-3}$, a significantly lower value than previously reported in the literature. We use planet interior models that assume a two-layer planet comprised of a liquid, pure Fe core and iron-free, $\rm MgSiO_{3}$ mantle, and we determine the range of core mass fractions that are consistent with the observed mass and radius. We use existing high-resolution spectra of the host star to derive Fe/Mg/Si abundances ([Fe/H]$=-0.03 \pm 0.01$, [Mg/H]$= 0.04 \pm 0.02$, [Si/H]$=0.03 \pm 0.06$) to infer the composition of K2-106b. We find that although K2-106b has a high density and core mass fraction ($44^{+12}_{-15}\%$) compared to the Earth ($33\%$), its composition is consistent with what is expected assuming that it reflects the relative refractory abundances of its host star. K2-106b is therefore unlikely to be a super-Mercury, as has been suggested in previous literature.

astro-ph.EP↗

arXiv2022

OGLE-2018-BLG-0799Lb: a $q \sim 2.7 \times 10^{-3}$ Planet with Spitzer Parallax

We report the discovery and analysis of a planet in the microlensing event OGLE-2018-BLG-0799. The planetary signal was observed by several ground-based telescopes, and the planet-host mass ratio is $q = (2.65 \pm 0.16) \times 10^{-3}$. The ground-based observations yield a constraint on the angular Einstein radius $θ_{\rm E}$, and the microlensing parallax vector $\vecπ_{\rm E}$, is strongly constrained by the Spitzer data. However, the 2019 Spitzer baseline data reveal systematics in the Spitzer photometry, so there is ambiguity in the magnitude of the parallax. In our preferred interpretation, a full Bayesian analysis using a Galactic model indicates that the planetary system is composed of an $M_{\rm planet} = 0.26_{-0.11}^{+0.22}~M_{J}$ planet orbiting an $M_{\rm host} = 0.093_{-0.038}^{+0.082}~M_{\odot}$, at a distance of $D_{\rm L} = 3.71_{-1.70}^{+3.24}$ kpc. An alternate interpretation of the data shifts the localization of the minima along the arc-shaped microlens parallax constraints. This, in turn, yields a more massive host with median mass of $0.13 {M_{\odot}}$ at a distance of 6.3 kpc. This analysis demonstrates the robustness of the osculating circles formalism, but shows that further investigation is needed to assess how systematics affect the specific localization of the microlens parallax vector and, consequently, the inferred physical parameters.

astro-ph.EP↗

arXiv2022

Systematic KMTNet Planetary Anomaly Search, Paper I: OGLE-2019-BLG-1053Lb, A Buried Terrestrial Planet

In order to exhume the buried signatures of "missing planetary caustics" in the KMTNet data, we conducted a systematic anomaly search to the residuals from point-source point-lens fits, based on a modified version of the KMTNet EventFinder algorithm. This search reveals the lowest mass-ratio planetary caustic to date in the microlensing event OGLE-2019-BLG-1053, for which the planetary signal had not been noticed before. The planetary system has a planet-host mass ratio of $q = (1.25 \pm 0.13) \times 10^{-5}$. A Bayesian analysis yields estimates of the mass of the host star, $M_{\rm host} = 0.61_{-0.24}^{+0.29}~M_\odot$, the mass of its planet, $M_{\rm planet} = 2.48_{-0.98}^{+1.19}~M_{\oplus}$, the projected planet-host separation, $a_\perp = 3.4_{-0.5}^{+0.5}$ au, and the lens distance of $D_{\rm L} = 6.8_{-0.9}^{+0.6}$ kpc. The discovery of this very low mass-ratio planet illustrates the utility of our method and opens a new window for a large and homogeneous sample to study the microlensing planet-host mass-ratio function down to $q \sim 10^{-5}$.

astro-ph.EP↗

arXiv2022

A Ubiquitous Unifying Degeneracy in Two-Body Microlensing Systems

While gravitational microlensing by planetary systems provides unique vistas on the properties of exoplanets, observations of a given 2-body microlensing event can often be interpreted with multiple distinct physical configurations. Such ambiguities are typically attributed to the close-wide and inner-outer types of degeneracies that arise from transformation invariances and symmetries of microlensing caustics. However, there remain unexplained inconsistencies between aforementioned theories and observations. Here, leveraging a fast machine learning inference framework, we present the discovery of the offset degeneracy, which concerns a magnification-matching behaviour on the lens-axis and is formulated independent of caustics. This offset degeneracy unifies the close-wide and inner-outer degeneracies, generalises to resonant topologies, and upon reanalysis, not only appears ubiquitous in previously published planetary events with 2-fold degenerate solutions, but also resolves prior inconsistencies. Our analysis demonstrates that degenerate caustics do not strictly result in degenerate magnifications and that the commonly invoked close-wide degeneracy essentially never arises in actual events. Moreover, it is shown that parameters in offset degenerate configurations are related by a simple expression. This suggests the existence of a deeper symmetry in the equations governing 2-body lenses than previously recognised.

astro-ph.EP↗