ABSTRACT
In previous work, we developed the idea to solve Kepler’s equation with a CORDIC-like algorithm, which does not require any division, but still requires multiplications in each iteration. ...Here we overcome this major shortcoming and solve Kepler’s equation using only bitshifts, additions and one initial multiplication. We prescale the initial vector with the eccentricity and the scale correction factor. The rotation direction is decided without correction for the changing scale. We find that double CORDIC iterations are self-correcting and compensate for possible wrong rotations in subsequent iterations. The algorithm needs 75 per cent more iterations and delivers the eccentric anomaly and its sine and cosine terms times the eccentricity. The algorithm can also be adopted for the hyperbolic case. The new shift-and-add algorithm brings Kepler’s equation close to hardware and allows it to be solved with cheap and simple hardware components.
Context. Many algorithms to solve Kepler’s equations require the evaluation of trigonometric or root functions. Aims. We present an algorithm to compute the eccentric anomaly and even its cosine and ...sine terms without usage of other transcendental functions at run-time. With slight modifications it is also applicable for the hyperbolic case. Methods. Based on the idea of CORDIC, our method requires only additions and multiplications and a short table. The table is independent of eccentricity and can be hardcoded. Its length depends on the desired precision. Results. The code is short. The convergence is linear for all mean anomalies and eccentricities e (including e = 1). As a stand-alone algorithm, single and double precision is obtained with 29 and 55 iterations, respectively. Half or two-thirds of the iterations can be saved in combination with Newton’s or Halley’s method at the cost of one division.
Ultra-hot Jupiters are emerging as a new class of exoplanets. Studying their chemical compositions and temperature structures will improve our understanding of their mass loss rate as well as their ...formation and evolution. We present the detection of ionized calcium in the two hottest giant exoplanets – KELT-9b and WASP-33b. By using transit datasets from CARMENES and HARPS-N observations, we achieved high-confidence-level detections of Ca II using the cross-correlation method. We further obtain the transmission spectra around the individual lines of the Ca II H&K doublet and the near-infrared triplet, and measure their line profiles. The Ca II H&K lines have an average line depth of 2.02 ± 0.17% (effective radius of 1.56 Rp) for WASP-33b and an average line depth of 0.78 ± 0.04% (effective radius of 1.47 Rp) for KELT-9b, which indicates that the absorptions are from very high upper-atmosphere layers close to the planetary Roche lobes. The observed Ca II lines are significantly deeper than the predicted values from the hydrostatic models. Such a discrepancy is probably a result of hydrodynamic outflow that transports a significant amount of Ca II into the upper atmosphere. The prominent Ca II detection with the lack of significant Ca I detection implies that calcium is mostly ionized in the upper atmospheres of the two planets.
High resolution transit spectroscopy has proven to be a reliable technique for the characterization of the chemical composition of exoplanet atmospheres. Taking advantage of the broad spectral ...coverage of the CARMENES spectrograph, we initiated a survey aimed at characterizing a broad range of planetary systems. Here, we report our observations of three transits of GJ 3470 b with CARMENES in search of He (2
3
S) absorption. On one of the nights, the He
I
region was heavily contaminated by OH
−
telluric emission and, thus, it was not useful for our purposes. The remaining two nights had a very different signal-to-noise ratio (S/N) due to weather. They both indicate the presence of He (2
3
S) absorption in the transmission spectrum of GJ 3470 b, although a statistically valid detection can only be claimed for the night with higher S/N. For that night, we retrieved a 1.5 ± 0.3% absorption depth, translating into a
R
p
(
λ
)∕
R
p
= 1.15 ± 0.14 at this wavelength. Spectro-photometric light curves for this same night also indicate the presence of extra absorption during the planetary transit with a consistent absorption depth. The He (2
3
S) absorption is modeled in detail using a radiative transfer code, and the results of our modeling efforts are compared to the observations. We find that the mass-loss rate,
Ṁ
, is confined to a range of 3 × 10
10
g s
−1
for
T
= 6000 K to 10 × 10
10
g s
−1
for
T
= 9000 K. We discuss the physical mechanisms and implications of the He
I
detection in GJ 3470 b and put it in context as compared to similar detections and non-detections in other Neptune-size planets. We also present improved stellar and planetary parameter determinations based on our visible and near-infrared observations.
Context.
The detection of Earth-mass exoplanets in the habitable zone around solar-mass stars using the radial velocity technique requires extremely high precision, on the order of 10 cm s
−1
. This ...puts the required noise floor below the intrinsic variability of even relatively inactive stars, such as the Sun. One such variable is convective blueshift varying temporally, spatially, and between spectral lines.
Aims.
We develop a new approach for measuring convective blueshift and determine the strength of convective blueshift for 810 stars observed by the HARPS spectrograph, spanning spectral types late-F, G, K, and early-M. We derive a model for infering blueshift velocity for lines of any depth in later-type stars of any effective temperature.
Methods.
Using a custom list of spectral lines, covering a wide range of absorption depths, we create a model for the line-core shift as a function of line depth, commonly known as the third signature of granulation. For this we utilize an extremely-high-resolution solar spectrum (
R
~ 1 000 000) to empirically account for the nonlinear nature of the third signature. The solar third signature is then scaled to all 810 stars. Through this we obtain a measure of the convective blueshift relative to the Sun as a function of stellar effective temperature.
Results.
We confirm the general correlation of increasing convective blueshift with effective temperature and establish a tight, cubic relation between the two that strongly increases for stars above ~5800 K. For stars between ~4100 and ~4700 K we show, for the first time, a plateau in convective shift and a possible onset of a plateau for stars above 6000 K. Stars below ~4000 K show neither blueshift nor redshift. We provide a table that lists expected blueshift velocities for each spectral subtype in the data set to quickly access the intrinsic noise floor through convective blueshift for the radial velocity technique.
Flat-relative optimal extraction Zechmeister, M; Anglada-Escude, G; Reiners, A
Astronomy and astrophysics (Berlin),
1/2014, Volume:
561
Journal Article
Peer reviewed
Open access
Context. Optimal extraction is a key step in processing the raw images of spectra as registered by two-dimensional detector arrays to a one-dimensional format. Previously reported algorithms ...reconstruct models for a mean one-dimensional spatial profile to assist a properly weighted extraction. Aims. We outline a simple optimal extraction algorithm (including error propagation), which is very suitable for stabilised, fibre-fed spectrographs and does not model the spatial profile shape. Methods. A high signal-to-noise ratio, master-flat image serves as reference image and is directly used as an extraction profile mask. Each extracted spectral value is the scaling factor relative to the cross-section of the unnormalised master flat that contains all information about the spatial profile, as well as pixel-to-pixel variations, fringing, and blaze. The extracted spectrum is measured relative to the flat spectrum. Results. Using echelle spectra of the HARPS spectrograph we demonstrate a competitive extraction performance in terms of a signal-to-noise ratio and show that extracted spectra can be used for high precision radial velocity measurement. Conclusions. Pre- or post-flat-fielding of the data is not necessary, since all spectrograph inefficiencies inherent to the extraction mask are automatically accounted for. Also the reconstruction of the mean spatial profile by models is not needed, thereby reducing the number of operations to extract spectra. Flat-relative optimal extraction is a simple, efficient, and robust method that can be applied easily to stabilised, fibre-fed spectrographs.
Aims. We explore the capabilities of CARMENES for characterising hot-Jupiter atmospheres by targeting multiple water bands, in particular, those at 1.15 and 1.4 μm. Hubble Space Telescope ...observations suggest that this wavelength region is relevant for distinguishing between hazy and/or cloudy and clear atmospheres. Methods. We observed one transit of the hot Jupiter HD 189733 b with CARMENES. Telluric and stellar absorption lines were removed using SYSREM, which performs a principal component analysis including proper error propagation. The residual spectra were analysed for water absorption with cross-correlation techniques using synthetic atmospheric absorption models. Results. We report a cross-correlation peak at a signal-to-noise ratio (S/N) of 6.6, revealing the presence of water in the transmission spectrum of HD 189733 b. The absorption signal appeared slightly blueshifted at –3.9 ± 1.3 km s−1. We measured the individual cross-correlation signals of the water bands at 1.15 and 1.4 μm, finding cross-correlation peaks at S/N of 4.9 and 4.4, respectively. The 1.4 μm feature is consistent with that observed with the Hubble Space Telescope. Conclusions. The water bands studied in this work have been mainly observed in a handful of planets from space. Being able also to detect them individually from the ground at higher spectral resolution can provide insightful information to constrain the properties of exoplanet atmospheres. Although the current multi-band detections can not yet constrain atmospheric haze models for HD 189733 b, future observations at higher S/N could provide an alternative way to achieve this aim.
Context.
Solar surface magneto-convection appears as granulation pattern that impacts spectral lines in terms of both shape and wavelength. Such induced effects also tend to vary over the observed ...solar disc because of the changing observation angle and, thus, the changing observation height as well. Centre-to-limb observations of the resolved Sun offer an insight into the variable spectral behaviour across different heliocentric observing positions, providing crucial information about limb darkening, convective velocities, and line profile variability relevant to radial velocity (RV) calculations. Thus, RV measurements and exoplanet transit spectroscopy depend on precise reference templates.
Aims.
We want to provide a spectroscopic centre-to-limb solar atlas at high spectral resolution and high-frequency accuracy. The atlas shall help improve the understanding of the solar atmosphere and convection processes.
Methods.
We performed high-resolution observations of the resolved quiet Sun with a Fourier transform spectrograph at the Institut für Astrophysik und Geophysik in Göttingen. Our dataset contains a wavelength range from 4200 Å to 8000 Å. We obtained 165 spectra in total, with a spectral resolution of Δ
ν
= 0.024 cm
−1
, corresponding to a resolving power
R
of 700 000 at ∼6000 Å.
Results.
We present a centre-to-limb solar atlas containing 14 heliocentric positions. To check for consistency, we investigated the Fe
I
6175 Å absorption line and compared our line profiles with previous centre-to-limb observations and also with simulations. The line profile and also the bisector profiles are generally consistent with previous observations, but we have identified differences to model line profiles, especially close to the solar limb.
Ultra-hot Jupiters (UHJs) are highly irradiated giant exoplanets with extremely high day-side temperatures, which lead to thermal dissociation of most molecular species. It is expected that the ...neutral hydrogen atom is one of the main species in the upper atmospheres of UHJs. Neutral hydrogen has been detected in several UHJs by observing their Balmer line absorption. In this work, we report four transit observations of the UHJ WASP-33b, performed with the CARMENES and HARPS-North spectrographs, and the detection of the H
α
, H
β
, and H
γ
lines in the planetary transmission spectrum. The combined H
α
transmission spectrum of the four transits has an absorption depth of 0.99 ± 0.05%, which corresponds to an effective radius of 1.31 ± 0.01
R
p
. The strong H
α
absorption indicates that the line probes the high-altitude thermosphere. We further fitted the three Balmer lines using the
PAWN
model, assuming that the atmosphere is hydrodynamic and in local thermodynamic equilibrium. We retrieved a thermosphere temperature 12 200
−1000
+1300
K and a mass-loss rate
Ṁ
= 1011.8
−0.5
+0.6
g s
−1
. The retrieved high mass-loss rate is compatible with the “Balmer-driven” atmospheric escape scenario, in which the stellar Balmer continua radiation in the near-ultraviolet is substantially absorbed by excited hydrogen atoms in the planetary thermosphere.
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