Wavelength calibration for a precision test of the fine-structure constant
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A&A 646, A144 (2021)
https://doi.org/10.1051/0004-6361/202039345 Astronomy
c ESO 2021 &
Astrophysics
Fundamental physics with Espresso: Towards an accurate
wavelength calibration for a precision test of the fine-structure
constant
Tobias M. Schmidt1,4 , Paolo Molaro1,2 , Michael T. Murphy3 , Christophe Lovis4 , Guido Cupani1,2 ,
Stefano Cristiani1,2 , Francesco A. Pepe4 , Rafael Rebolo5,6 , Nuno C. Santos7,8 , Manuel Abreu9,10 ,
Vardan Adibekyan7,11,8 , Yann Alibert12 , Matteo Aliverti13 , Romain Allart4 , Carlos Allende Prieto5,6 , David Alves9,10 ,
Veronica Baldini14 , Christopher Broeg15 , Alexandre Cabral9,10 , Giorgio Calderone1 , Roberto Cirami1 ,
João Coelho9,10 , Igor Coretti1 , Valentina D’Odorico1,2 , Paolo Di Marcantonio1 , David Ehrenreich4 ,
Pedro Figueira7,16 , Matteo Genoni13 , Ricardo Génova Santos5,6 , Jonay I. González Hernández5,6 , Florian Kerber17 ,
Marco Landoni18,13 , Ana C. O. Leite7,8 , Jean-Louis Lizon17 , Gaspare Lo Curto17 , Antonio Manescau17 ,
Carlos J. A. P. Martins7,11 , Denis Megévand4 , Andrea Mehner16 , Giuseppina Micela19 , Andrea Modigliani17 ,
Manuel Monteiro7 , Mario J. P. F. G. Monteiro7,8 , Eric Mueller17 , Nelson J. Nunes9,10 , Luca Oggioni13 ,
António Oliveira9,10 , Giorgio Pariani13 , Luca Pasquini17 , Edoardo Redaelli13 , Marco Riva13 , Pedro Santos9,10 ,
Danuta Sosnowska4 , Sérgio G. Sousa7 , Alessandro Sozzetti20 , Alejandro Suárez Mascareño5,6 , Stéphane Udry4 ,
Maria-Rosa Zapatero Osorio21 , and Filippo Zerbi13
(Affiliations can be found after the references)
Received 4 September 2020 / Accepted 18 November 2020
ABSTRACT
Observations of metal absorption systems in the spectra of distant quasars allow one to constrain a possible variation of the fine-structure constant
throughout the history of the Universe. Such a test poses utmost demands on the wavelength accuracy and previous studies were limited by
systematics in the spectrograph wavelength calibration. A substantial advance in the field is therefore expected from the new ultra-stable high-
resolution spectrograph Espresso, which was recently installed at the VLT. In preparation of the fundamental physics related part of the Espresso
GTO program, we present a thorough assessment of the Espresso wavelength accuracy and identify possible systematics at each of the different
steps involved in the wavelength calibration process. Most importantly, we compare the default wavelength solution, which is based on the
combination of Thorium-Argon arc lamp spectra and a Fabry-Pérot interferometer, to the fully independent calibration obtained from a laser
frequency comb. We find wavelength-dependent discrepancies of up to 24 m s−1 . This substantially exceeds the photon noise and highlights the
presence of different sources of systematics, which we characterize in detail as part of this study. Nevertheless, our study demonstrates the
outstanding accuracy of Espresso with respect to previously used spectrographs and we show that constraints of a relative change of the fine-
structure constant at the 10−6 level can be obtained with Espresso without being limited by wavelength calibration systematics.
Key words. instrumentation: spectrographs – techniques: spectroscopic – cosmology: observations
1. Introduction would shift the wavelengths of spectral lines and is therefore
observable. The strength of the effect depends on the electron
The mathematical1 description of the phenomena of Nature, for- configuration of an atom and different spectral lines can have a
malized as our laws of physics, requires a set of fundamental con- substantially different sensitivity to the value of the fine-structure
stants (e.g., G, ~, c, ...), which determine the scales of physical constant (e.g., Dzuba et al. 1999). By observing multiple transi-
effects. The values of these constants cannot be predicted by the- tions originating from the same absorption system, one can break
ory but have to be determined experimentally. In generalized field the degeneracy between absorption redshift and fine-structure
theories that try to relate physical constants to more fundamen- constant and therefore directly constrain the value of α at the
tal concepts, these constants can in principle depend on time and time and place where the absorption happened. Thus, the con-
space (see e.g., Uzan 2011). Fortunately, such a possible change of straint on α basically comes from an accurate measurement of
fundamental constants can be tested, for instance, by ultra-precise relative wavelength differences.
laboratory experiments on Earth (e.g., Rosenband 2008), but also The classical approach for such experiments is to study
with astronomical observations in the distant Universe. metal absorption lines in the spectra of distant quasars (e.g.,
In the past, most attention has been directed towards the fine- Wolfe et al. 1976; Webb et al. 1999). Figure 1 illustrates the gen-
2
structure constant α = 4πe0 ~ c ≈ 137
1
, which defines the coupling eral concept of such an observation and indicates the amplitude
strength of electromagnetic interactions and therefore affects the of the wavelength shift for a few commonly used metal transi-
energy levels of atomic transitions. In practice, a change of α tions. Among the absorption lines with the strongest shift are the
Fe ii lines with rest wavelengths between 2300 Å and 2600 Å.
1
Based on work by the fundamental constants working group of the They shift by about 20 m s−1 for a relative change of α of one
ESPRESSO GTO Consortium. part in a million (ppm; see e.g., Murphy & Berengut 2014). This
Article published by EDP Sciences A144, page 1 of 26A&A 646, A144 (2021)
These studies confirm intra-order distortions of 300 m s−1 PtV
∆α / α = +5.00 ppm × 105
3.0 and global slopes up to 800 m s−1 over 1500 Å for UVES and
SiII AlII AlIII FeII FeII MgI 600 m s−1 over 3000 Å for HIRES (Whitmore & Murphy 2015).
2.5
FeII AlIII FeII FeII MgII The wavelength calibration of HARPS (Mayor et al. 2003) is
2.0 much more accurate (Whitmore & Murphy 2015; Cersullo et al.
Flux
2019), however, installed at the ESO 3.6 m telescope, this
1.5
spectrograph is simply not equipped with sufficient collecting
1.0 area to observe any except the brightest quasar suitable for this
experiment (Kotuš et al. 2017; Milakovic et al. 2020a).
0.5
A substantial advance of this field is therefore expected
0.0 from the new ultra-stable high-resolution Echelle SPectro-
4000 4500 5000 5500 6000 6500 7000 7500
graph for Rocky Exoplanets and Stable Spectroscopic Observa-
Wavelength in Å
tions (Espresso, Molaro 2009; Pepe et al. 2010, 2014, 2021;
Fig. 1. Illustration of the general concept for measuring a variation of Mégevand et al. 2014), recently installed at the VLT. Espresso
the fine-structure constant from a fiducial metal absorption systems at was designed for ultra-precise radial-velocity (RV) studies of exo-
zabs = 1.7 in a quasar spectrum. The assumed change of ∆α/α by 5 ppm planets and for a precision test of the variability of fundamental
leads to differential line shifts, indicated by the arrows. For visibility, constants.
the magnitude of the shifts is exaggerated by ×105 . Espresso is located at the incoherently-combined Coudé
focus of the VLT and can in 1-UT mode be fed by any of the
four 8.2 m mirrors or in the 4-UT mode with the (incoherently)
has to be compared to lines with low sensitivity to α, often from combined light of all four VLT telescopes2 . Several design aspects
light elements such as Mg ii ('3 m s−1 shift per ppm) or Al iii are essential for its stability and accuracy. First, the spectrograph
('4 m s−1 per ppm). A somehow special transition is the Fe ii is located inside a thermally controlled (∆T ' 1 mK) vacuum
λ = 1608 Å line, which has a relatively strong dependence on α vessel and therefore not affected by environmental effects like air
but with opposite sign (−12 m s−1 per ppm). It is therefore ideal temperature and pressure, which influence the index of refrac-
to compare it with the other Fe ii lines, but, unfortunately, its tion and would therefore change the echellogram. Second, it has
oscillator strength is relatively low and the line is not always zero movable components within the spectrograph vessel and
observed at a sufficient strength and signal-to-noise ratio. thus a fixed spectral format. Third, Espresso is fed by optical
The current constraints reported on ∆α/α, either from large fibers3 , facilitating a much more stable illumination of the spec-
ensembles of absorbers (e.g., Murphy et al. 2003; Chand et al. trograph than possible with slit spectrographs. A high degree of
2004; King et al. 2012) or individual absorption systems immunity to changes of the fiber input illumination is achieved
(Levshakov et al. 2007; Molaro et al. 2008a, 2013a; Kotuš et al. by the use of octagonal fibers, which have a high (static) mode-
2017), have a claimed precision between one and a few ppm. scrambling efficiency and provide a very homogeneous output
However, there is disagreement among these studies whether profile (Chazelas et al. 2012). In addition, the near- and far-field
the fine-structure constant actually varies or not and to which of the fibers are exchanged between two sections of the fiber train,
degree the measurements are affected by (instrumental) system- a design described as double-scrambler (e.g., Hunter & Ramsey
atics. Since all these studies are working close to the instrument’s 1992; Bouchy et al. 2013). To minimize guiding errors and ensure
wavelength accuracy, new measurements with more accurate an optimal injection of light into the optical fibers, active field and
spectrograps are clearly needed to settle the issue. These have to pupil stabilization by the means of guide cameras and piezo tip-tilt
deliver a precision of at least 1 ppm to provide meaningful con- mirrors is incorporated in the spectrograph front-end (Riva et al.
straints, which poses utmost demands on the wavelength calibra- 2014; Calderone et al. 2016; Landoni et al. 2016). In addition,
tion of the spectrograph. The line shifts stated above imply that atmospheric dispersion correctors (ADC) compensate the effect
the utilized spectrograph shall not exhibit peak-to-valley (PtV) of differential refraction in Earth’s atmosphere.
distortions of the wavelength scale that are substantially larger These measures guarantee an extremely stable and homoge-
than '20 m s−1 to safely exclude wavelength calibration uncer- neous illumination of the spectrograph and therefore eliminate
tainties. Achieving this goal is extremely challenging. several issues that have plagued previous studies of a possible
It turned out that the existing high-resolution echelle spec- change of the fine-structure constant, namely slit and pupil illu-
trographs Keck/HIRES (Vogt et al. 1994) and VLT/UVES mination effects due to guiding errors, placement of the target
(Dekker et al. 2000) are neither designed nor ideal for this task. on the slit, variable seeing, and atmospheric dispersion effects4
For instance, Griest et al. (2010) and Whitmore et al. (2010)
demonstrated by observations taken through iodine cells that the 2
However, in the 4-UT mode (4MR) only at medium resolution of
standard wavelength calibrations of HIRES and UVES, derived R ≈ 72 000, approximately half of the resolution possible in the high-
from exposures of thorium-argon (ThAr) hollow-cathode lamps resolution 1-UT mode (1HR), which offers R ≈ 138 000.
3
(HCL), show intra-order distortions of up to 700 m s−1 PtV and To facilitate high optical throughput, the light from the individual UTs
overall shifts between exposures up to ±1000 m s−1 . This raised is guided by classical optical trains composed of mirrors, prisms and
strong concerns if the '1 ppm constraints on the fine-structure lenses to the combined Coudé laboratory where the light is injected into
relatively short optical fibers.
constant derived with these instruments are trustworthy. More 4
For Espresso, imperfect placement of the target image on the fiber,
recently, attempts have been made to supercalibrate quasar spec- for example due to guiding errors or incomplete correction of differ-
tra (Molaro et al. 2008b; Rahmani et al. 2013; Evans et al. 2014; ential atmospheric refraction, would lead to increased (differential) slit
Whitmore & Murphy 2015; Murphy & Cooksey 2017), which 1
losses but not affect spectral features. Due to the λ− 4 dependence of
intend to transfer the wavelength accuracy of Fourier-transform atmospheric seeing, slit losses are anyway wavelength dependent. This
spectrometers at solar observatories (e.g., Reiners et al. 2016) to might complicate spectrophotometric flux calibration but the sophisti-
the quasar observations by using spectra of solar light reflected cated mode scrambling in the fiber feed ensures that the inferred wave-
off asteroids or solar twins, which are stars very similar to the Sun. lengths of spectral features remain unaffected.
A144, page 2 of 26T. M. Schmidt et al.: Towards an accurate Espresso wavelength calibration
(e.g., Murphy et al. 2001). In addition, the fiber feed also elimi- in particular to a test for a possible variation of the fine-structure
nates noncommon path aberrations between science target and constant.
wavelength calibration sources since in both cases the light However, the claimed wavelength accuracy has to be demon-
passes through exactly the same optical path5 . strated. The goal of this study is therefore a careful and thor-
To facilitate an accurate wavelength calibration, Espresso ough assessment of the Espresso wavelength calibration. It has
is equipped with a comprehensive suite of calibration sources to be stressed that the calibration requirements for tests of a
(Mégevand et al. 2014). This encompasses a classical ThAr hol- varying fine-structure constant are fundamentally different from
low cathode lamp, which is complemented by a Fabry-Pérot those of RV studies aimed at detecting and characterizing exo-
interferometer that produces an extremely large number of nar- planets. Radial-velocity studies require extreme precision and
row (but marginally resolved), equally spaced lines (Wildi et al. repeatability since the signal is the difference between observa-
2010, 2011, 2012; Bauer et al. 2015). Although not providing tions taken with the same instrument at different times. Also, the
absolute wavelength information, the Fabry-Pérot interferome- aim is to measure a global RV shift for which all (or a selec-
ter allows in combination with the ThAr lamp a much more tion of) absorption lines across the spectral range are combined.
precise calibration, in particular on small scales, as would be Distortions of the wavelength scale are therefore of little impor-
possible with arc spectra alone (demonstrated for HARPS by tance, at least if they remain stable. This is different for a test
Cersullo et al. 2019). of fundamental physics. Stability of the instrument is clearly
A completely independent means of wavelength calibration6 highly convenient, but it is technically not essential since the
comes from a laser frequency comb (LFC). The core element constraint comes from the wavelength difference between dif-
of the LFC is a passively mode-coupled femto-second laser ferent absorption lines within the same observation. Thus, it is
that provides a dense train of extremely narrow and equally- crucial to have an accurate wavelength scale that is free of dis-
spaced emission lines, which frequencies are directly stabi- tortions. In fact, a global RV shift would be the only thing not
lized against a (local or remote) atomic clock and therefore the relevant for a constraint of the fine-structure constant since it is
fundamental SI time standard (e.g., Murphy et al. 2007, 2012; degenerate with the absorption redshift of the system and there-
Steinmetz et al. 2008; Wilken et al. 2010a,b, 2012; Probst et al. fore unimportant. However, this is only true if the shift is con-
2014). With intrinsic accuracies reaching 10−12 , LFCs promise stant in velocity space. A shift constant in wavelength, frequency
to deliver unprecedented calibration accuracies for astronomical or pixel position would cause a wavelength-dependent velocity
spectrographs. This novel calibration method was in the opti- shift and therefore not cancel out.
cal first applied to HARPS (Wilken et al. 2010a; Coffinet et al. To assess the Espresso wavelength accuracy, we present
2019; Cersullo et al. 2019). One application is for example the here all steps required to obtain the wavelength solution of the
definition of a solar atlas (Molaro et al. 2013b), obtained from spectrograph, starting from the basic data reduction (Sect. 2)
LFC-calibrated asteroid spectra. A solar spectrum calibrated in and spectral extraction (Sect. 3), fitting of the calibration source
this way can then, again via asteroid (or solar twin) observa- spectra (Sect. 4.1) and finally the computation of the actual
tions, be used to calibrate quasar spectra taken at larger tele- wavelength solution (Sects. 4.3 and 4.4). While going along, we
scopes. This is similar to the supercalibration technique by identify issues that might lead to systematics for constraining
Whitmore & Murphy (2015) but uses directly the absolute wave- α and wherever possible carry out consistency checks to test
length information provided by the LFC instead of relying and demonstrate the accuracy of the derived results. The most
on observations taken with solar Fourier-transform spectrome- informative test is the comparison of the wavelength solution
ter, which by itself are calibrated by further secondary means. obtained jointly from the ThAr arc lamp spectra and the Fabry-
Espresso should in the best case render these complex proce- Pérot interferometer to the one derived from the laser frequency
dures obsolete. Since it is itself equipped with a LFC, no extra comb (Sect. 5). The comparison of these two fully independent
steps are required to achieve the same calibration accuracy. All solutions gives a clear picture of the wavelength accuracy and
these measures should make Espresso the astronomical spec- allows to predict the impact of systematic effects for a test of
trograph that achieves the highest wavelength accuracy, only changing fine-structure constant (Sect. 6.3).
rivaled by Fourier-transform spectrometers at solar observatories
or laboratories.
Despite the complex Coudé train, the fiber feed, the 2. Basic data reduction
extremely high resolution, and numerous extra steps taken to The results presented in this study are based on a set of custom-
ensure stability and accuracy, Espresso is designed for high developed routines for data reduction and wavelength calibra-
efficiency and offers (arguably, and depending on the exact tion. We therefore do not make use of the Espresso Data
configuration and conditions) a similar throughput as UVES Reduction Software (DRS, Lovis et al., in prep.), which is
(Pepe et al. 2021), but at much higher resolution (R = 138 000 the standard ESO instrument pipeline for this spectrograph7 .
compared to R ≈ 50 000 using a 0.8" slit) and with larger The reasons for this approach are manifold. First, the DRS is
instantaneous wavelength coverage (from 3840 Å to 7900 Å). designed as robust general-purpose instrument pipeline and opti-
It is therefore ideally suited for precision tests of fundamental mized for RV studies for exoplanet research. As described above,
physics and, logically, the Espresso consortium has dedicated the requirements for a precision test of fundamental constants
10% of the GTO time to fundamental physics related projects, are different from those of RV studies. Our code therefore fol-
5
The relevant procedure for fine-structure studies is that, before or lows a design philosophy that is optimized for accuracy instead
after the science observation, light from the calibration source(s) is fed of precision. In particular high priority was given to the mini-
through the same fiber as the light from the science target. This is unre- mization of correlations of wavelength errors across the spectral
lated to the simultaneous reference method used in RV studies. range. The overall wavelength calibration strategy is of course
6 governed by the instrument design and calibration scheme.
The LFC calibration requires an initial wavelength solution for line
identification. However, the requirements for this are moderate and the
7
final LFC solution is formally independent of the a-priori assumed https://www.eso.org/sci/software/pipelines/espresso/
wavelength solution. Details are given in Sect. 4.4. espresso-pipe-recipes.html
A144, page 3 of 26A&A 646, A144 (2021)
Therefore, our code employs similar procedures than the DRS. only single frames are available per calibration session. These
Still, we try to optimize towards our science case, which results are just overscan and masterbias corrected. No flatfielding, nei-
in the use of different algorithms for several tasks. For exam- ther by chip flatfields nor spectral flatfields is applied. Pixel-to-
ple, we try to wherever possible approximate functions locally pixel sensitivity variations will be accounted for as part of the
by nonparametric methods instead of fitting global polynomials. spectral extraction process. Also, no masking of bad pixels is
These techniques should deliver better wavelength accuracy at applied since this is not needed within the context of this study.
the expense of precision, which is not the limiting factor for a test
of fundamental constants. Apart from this, a fully independent
implementation allows for cross-check between the pipelines 3. Spectral extraction
and allows to exclude systematics caused by the implementa- Following this basic reduction of the wavelength calibration
tion itself (presented in Sect. 6.1). But most notably, our code- frames, the next step is the spectral extraction, specifically, the
base is intended as a flexible testbed to experiment and develop gathering of spectral information from the two-dimensional raw
improved algorithms, without the need to obey the strict require- frames and their reduction to one-dimensional spectra. This is
ments of official ESO instrument pipelines. a crucial aspect of the data reduction procedure and therefore
All data presented here were taken as part of the standard described in detail.
daily Espresso calibration plan on August 31 2019, within In general, extraction is done independently for each order.
a period of '2.5 h. Instrumental drifts over this timespans are Since Espresso is fed by two Fibers (A and B) and utilizes a
2 m s−1 and therefore negligible. Since the test of fundamental pupil slicer design that splits the pupil image in half to create
constants is in principle a single shot experiment that does not two images of the same fiber (Slice a and b) on the focal plane,
rely on a monitoring campaign, the assessment of the stability or four individual spectra per order are imaged onto the detectors.
time evolution is beyond the scope of this paper. We focus on the The extraction process should make use of optimal extrac-
1-UT, high-resolution mode of Espresso and if not stated oth- tion, a scheme initially described by Horne (1986) and now
erwise, the detectors were read using a 1 × 1 binning (1HR1x1). commonly used. As demonstrated in Horne (1986), the outlined
However, due to the faintness of the background quasars, all scheme is flux complete in the sense that it is able to deliver
observations for the fundamental physics project will utilize the an estimate of the total received flux and therefore spectropho-
1HR2x1 mode8 , which employs a 2× binning in cross-dispersion tometric results. In addition, it is optimal in the sense that it
direction. Therefore, we also carried out the full analysis for this minimizes the variance of the extracted spectrum. However, it
instrument mode and wherever necessary also present the results makes the fundamental assumption that there is a unique cor-
for the 1HR2x1 mode and state this explicitly. respondence between wavelength and pixel position, neglect-
The basic data reduction of the raw frames follows a stan- ing the finite extent of the instrumental point-spread function
dard procedure and is very similar to the DRS. Every frame is (PSF) in dispersion direction. Under this assumption, the dis-
first processed in the following way: for each of the 16 read- tribution of flux on the detector becomes a separable function
out amplifiers per detector, the bias value is determined from the of the spectral energy distribution of the source (SED) and the
overscan region and subtracted from the data. In the same step, profile of the trace in spatial (or cross-dispersion) direction. For
the overscan region is cropped away and ADUs converted to e− a detailed discussion see Zechmeister et al. (2014), in partic-
using the gain values listed in the fits headers (1.099 e− /ADU ular their Eqs. (2)–(4). This assumption drastically simplifies
and 1.149 e− /ADU for blue and red detector). This step also the extraction process, which is therefore mostly related to the
allows to determine the readout noise (RON), which is, depend- description of the (spatial) trace profile, which encapsulates the
ing on the amplifier, in the 1HR1x1 readout mode between 8.7 relative intensity of different pixels corresponding to the same
and 12.1 e− for the blue detector and between 6.8 and 14.2 e− in wavelength.
the red detector. The corresponding values in 1HR2x1 mode are Because Espresso is a fiber-fed echelle spectrograph, there
3.0–4.7 e− and 2.2–2.4 e− . The RON is stored separately for each is no spatial direction and the spectra projected on the detec-
output amplifier and propagated throughout the full analysis. tors contain no scientific information in cross-dispersion direc-
Following this, the ten bias frames taken as part of the daily tion. Since the echellogram is approximately aligned with the
Espresso calibration scheme are stacked and outliers identified pixel grid and the individual traces never tilted by more than
to reject cosmic ray hits, which are present even in bias frames. 10 degrees with respect to the detector coordinate, the spec-
The masked frames are then mean combined to form the master- tral geometry is approximated by assuming that the wavelength
bias, which encapsulates the spatial variation of the bias value. direction coincides with the detector Y direction and the cross-
The structure in the masterbias is rather small ('±4 e− ), except dispersion direction with the detector X direction. Therefore, the
in the corners of the individual chip readout areas. Still, we sub- trace profile Pyx at pixel position y is just a cut through the trace
tract the masterbias from all subsequent frames to correct for along the detector X coordinate.
this effect. Since the dark current is low ('1 e− h−1 pix−1 ) and Formally, the extraction process can be described as follows:
the exposure times at maximum 40 s, no dark current correction For each detector position (y|x), the bias and background sub-
is applied. tracted raw electron counts Cyx are divided by the normalized9
According to the Espresso calibration plan, every day two trace profile Pyx , which delivers for every pixel an independent
sets of ten spectral flatfield frames are taken in which always estimate of the total detected number of photons in the given
only one of the two fibers is illuminated. The two sets are treated detector column. These can then be averaged in cross-dispersion
independently, overscan corrected, converted to e− , the master- direction using inverse-variance weighting ensuring that pixels
bias subtracted and then stacked including an outlier detection to
9
reject cosmic ray hits. We emphasize that in contrastP to the formalism in Zechmeister et al.
For the wavelength calibration frames (Thorium-Argon arc (2014), Pyx is normalized, i.e., x Pyx = 1. This is desired, so that C̄y in
lamp, Fabry-Pérot Interferometer and laser frequency comb) Eq. (1) represents
q the physical number of detected photons and its error
thus be ' C̄y . In consequence, the blaze and spectral fluxing has to (or
8
Or in the future the 1HR4x2 heavy-binning mode. can) be determined separately.
A144, page 4 of 26T. M. Schmidt et al.: Towards an accurate Espresso wavelength calibration
in the center of the trace, which receive more flux and therefore regions and then interpolating these measurements using bivari-
show smaller relative errors, are weighted stronger than pixels ate splines of third order. In an iterative sigma-clipping process,
in the wings of the trace. The estimate for the total number of the traces themselves are masked and the background model
photons received in a certain detector column C̄y is therefore refined using only the intra-order regions, which is then sub-
tracted from the master flatfield. To define the trace profile Pyx , a
x C yx /Pyx × Wyx
P
region of ±∆xpr pixels around the trace center11 is extracted from
C̄y = . (1)
the master flatfield. As stated before, the exact extent of this win-
P
x Wyx
dow is of no particular importance, neither is the assumed center
By choosing weights Wyx according to the inverse variance of of the trace. The only requirement is that the trace is fully cov-
the measurements Cyx one obtains, as demonstrated in Horne ered by the assumed region.
(1986), an unbiased estimate of the true received flux with mini- As described in Eqs. (1) and (2), the trace profile Pyx deter-
mal variance. The variance for the individual pixels can be com- mines which pixels get extracted and therefore has to be zero
puted as far away from the trace center. Since the model is empirical
and taken from the master flatfield, the profile will inevitably
P2yx P2yx become noisy in the wings. This by itself should not be an issue
Wyx = = , (2) since the noise in the to-be-extracted spectrum should always
var(Cyx ) Cyx + BGyx + DARKyx + RONyx
2
dominate over the noise in the stacked master flatfield. However,
where BGyx and DARKyx are the contribution of scattered light imperfections in the scattered light subtraction might prevent the
and dark counts, while RONyx is the read-out noise. The com- trace profile to approach zero and therefore lead to biases. It was
plete extraction procedure is therefore solely governed by the thus decided to artificially truncate the trace profile. This is done
win
trace profile Pyx . by introducing a window function Wyx that acts as additional
weighting. Equation (1) therefore becomes
x C yx /Pyx × Wyx × Wyx
3.1. Determining the trace profile
P var win
C̄y = P var win
, (3)
It is common practice to model the trace profile with analytic x Wyx × Wyx
functions like a Gaussian and assume its properties (e.g., cen-
var win
tral position and width) to evolve slowly with detector position. with Wyx as defined in Eq. (2). For the window function Wyx ,
However, for Espresso the trace profile is non-Gaussian and fractional pixels are extracted, meaning that the window function
can not easily be described by analytic functions in an accu- is nonbinary12 and for an individual pixel proportional to the area
rate and precise way. Fortunately, since Espresso is fiber-fed of the pixels that lies within a region of ±∆xwin pixels around the
and designed for extreme stability, the spectral format is entirely trace center X0 (y). Formally, this can be written as
fixed and does not change with time. We therefore follow the Z xhi
approach described in Zechmeister et al. (2014), which uses an Wyx =
win
Θ(x0 − X0 (y) + ∆xwin )Θ( −x0 + X0 (y)
empirical model of the trace profile that can be directly derived xlo
from the master flatfield frame. In this way, the extraction is + ∆xwin )dx0 , (4)
applied completely blind and fully independent of the science
spectrum, purely defined by the trace profile observed in the with xlo and xhi the lower and upper bounds of the pixel at posi-
master flatfield. tion y and Θ(x0 ) the Heaviside step function. The size ∆xwin is
To extract the trace profile from the master flatfield frame chosen so that for example '95% of the flux is extracted13 .
one has to know its location. This requires an initial fit to the Introducing the window function now makes the extraction
trace profile, which, however, does not have to be particularly formally dependent on the initial determination of the trace cen-
accurate. As described in Eq. (1), the scaling and weighting of ter. However, the detected counts normalized by the trace pro-
individual pixels is fully defined by the trace profile and thus file should be constant along the trace profile, or in other words,
the observed flux in the master flatfield frame. It is therefore the quantity Cyx /Pyx should show no dependence on detector X
completely independent of the initial trace center as long as the coordinate. Therefore, shifting or in other ways modifying the
full extent of the trace is captured within an appropriately cho- window might slightly change the variance of the extracted spec-
sen window around the fitted trace center10 . This substantially trum but the flux estimate C̄y should remain unchanged. This is
relaxes the requirement on the accuracy of the initial tracing illustrated in Fig. 2, which shows a 2D representation of Cyx /Pyx
and allows us to follow a rather simple approach. One could fit in detector coordinates.
the trace profile for each detector Y position individually. How-
ever, given approx. 9000 pixels in detector Y direction and 340
traces, this poses a significant computational effort. To speed-up 3.2. Ambiguity of the trace profile
the computation, the data is binned by 16 pixels in detector Y Many aspects of the extraction process, in particular, being opti-
direction and then fitted by a simple Gaussian plus a constant mal in the sense that it minimizes the variance of the extracted
background. Based on the central values obtained by these fits, spectrum and the independence from the initial estimate of the
an estimate of the trace center for every pixel in detector Y direc-
tion X0 (y) is obtained by cubic spline interpolation. 11
In practice, ∆xpr = 10.5 pixels is used in 1HR1x1 mode and ∆xpr =
Following this, a global model of the scattered light is com- 5.5 pixels in 1HR2x1.
puted by first determining the median counts in 128 pix×128 pix 12 win
If Wyx would be a binary mask, it could be combined with Pyx in
one function (as e.g., in Zechmeister et al. 2014). The need to account
10 for fractional pixels requires to have two independent objects for the
One further requirement is that the empirical trace profile drops to
zero far away from the trace center. Since the calibration frames contain profile and the extraction window.
a significant amount of scattered light, an adequate procedure to subtract 13
Here, ∆xwin = 5.0 pixels is used in 1HR1x1 and ∆xwin = 2.5 pixels in
this background is needed. 1HR2x1 mode.
A144, page 5 of 26A&A 646, A144 (2021)
Rescaled Counts
0 150000 300000 450000
Detector X Coordinate
385
380
375
370
365
360
355
350
5400 5450 5500 5550 5600 5650
Detector Y Coordinate
Fig. 2. Small part of a single bias and scattered-light subtracted flatfield frame normalized by the trace profile. The pink dashed line indicates the
estimated trace center. As expected, there is no structure in vertical direction. The only visible effect is increased noise towards the upper and lower
edge of the trace. The vertical striping represents the individual pixel sensitivities but is mostly related to bad pixels. The striping, i.e., variations
in dispersion direction, is of no concern since it will be removed by the de-blazing and flux calibration procedures.
Rescaled Counts
0 60000 120000 180000 240000
Detector X Coordinate
385
380
375
370
365
360
355
350
5400 5450 5500 5550 5600 5650
Detector Y Coordinate
Fig. 3. Extraction of a small part of a Fabry-Pérot calibration frame. The plot shows the bias and scattered-light subtracted raw counts normalized
by the trace profile. The region on the detector is identical to the one shown in Fig. 2. Clearly visible is the sequence of equally-spaced narrow
emission lines. However, in contrast to the expectation, significant structure in cross-dispersion direction is present. The vertical purple lines
correspond to the positions of the cuts shown in Fig. 4.
trace center, are only valid as long as the assumed trace pro- In cross-dispersion direction, the FP lines basically resemble a
file Pyx is correct. As soon as the assumed profile deviates from double-horn profile. In addition, the flux in the wings of the trace
the true one, this no longer holds and the extracted spectrum profile does not show the clear pattern of sharp emission lines but
becomes imperfect. A slight increase of the variance in the out- only some very limited modulation, staying far below the line flux
put spectrum is acceptable to a certain degree, but the main measured in the core of the trace. A similar but inverted behavior is
concern within the context of a precision test of fundamental seen in between the FP lines. Here, one finds lower flux in the trace
constants are biases in the extraction process that later introduce center compared to its wings. The data obviously does not match
systematics in the wavelength solution. the expectations and clearly shows that the trace profile extracted
General shortcomings of the traditional optimal extraction from the flatfield frames is not capable of adequately describing
process have been described in detail by Bolton & Schlegel the trace profile observed in FP frames. Since normalizing the
(2010). However, due to the complexity of the issue, we demon- observed counts by the trace profile does not lead to a constant
strate in the following the particular effects for Espresso. For flux estimate across the trace, the averaging process described in
this, we focus on the extraction of a Fabry-Pérot frame used Eq. (3) will not yield an unbiased estimate of the true flux and thus
for wavelength calibration. The Fabry-Pérot interferometer (FP) depends on the choice of the extraction window.
produces a train of narrow (marginally resolved at R ≈ 138 000), The flatfield and FP frames were taken on the same day within
densely spaced (≈1.96 × 1010 Hz separation, corresponding to less than one hour. Given the exquisite stability of Espresso,
0.1 Å–0.4 Å) and equally bright lines and is therefore well-suited instrumental drifts can be ruled out. The imperfect subtraction of
for wavelength calibration purposes. The high number of equally scattered light could lead to a mismatch between the two trace pro-
spaced lines allows a very precise determination of the local files. However, we confirmed by not applying any scattered light
wavelength solution, far better than possible with the sparse, subtraction at all that this is not the dominant effect driving the
unevenly distributed lines produced by the classical ThAr hol- discrepancy seen in the extraction of the FP frame.
low cathode arc lamps. Figure 3 also shows that the FP lines are slightly tilted with
Figure 3 shows a small region of a FP spectrum in detec- respect to the detector coordinates. This tilt is small and corre-
tor coordinates. Similar to Fig. 2, the raw counts are bias and sponds only to a fraction of a pixel difference between upper
scattered-light corrected and divided by the trace profile. Clearly and lower part of the trace but still might be relevant. The final
visible is the series of narrow emission lines produced by the wavelength solution is required to be accurate to better than
FP interferometer. However, in stark contrast to the flatfield '20 m s−1 , corresponding to only 4% of a pixel. Thus, even very
frame, the FP frame exhibits very pronounced structure in cross- minute effects can impact the data products at a very significant
dispersion direction. Most importantly, the flux estimate for level. Due to the tilt of the lines and the need of an explicit trun-
the lines is lower in the center of the trace compared to posi- cation of the trace profile, the initial fit to the trace center could
tions approximately five pixels above and below the center. have an impact on the final wavelength solution.
A144, page 6 of 26T. M. Schmidt et al.: Towards an accurate Espresso wavelength calibration
Red Arm, Order 105, Fiber A, Slice a, Y = 5518 ± 4
30000 0.14
0.12
25000
Normalized Flux
0.10
20000
Counts
0.08
15000
0.06
10000
0.04
5000 0.02
0 0.00
6910 6915 6920 6925 6910 6915 6920 6925
0.12 300000
0.10 250000
Extraction Profile
Extracted Counts
0.08 200000
0.06 150000
0.04 100000
0.02 50000
0.00 0
6910 6915 6920 6925 6910 6915 6920 6925
Detector X Coordinate Detector X Coordinate
Fig. 4. Illustration of various cuts through the trace in detector X direction (cross-dispersion). Top left panel: nine trace profiles shown in terms
of raw counts, i.e., Cyx . The central cut lies exactly at the peak of a FP line. The others are offset in detector Y (wavelength) direction by up to
±4 pixels. The saturation of the line color indicates the offset from the line. Top right panel: same as top-left panel but all profiles are normalized
to unity flux, highlighting the different shapes. Bottom left panel: trace profiles extracted from a spectral flatfield frame at the same positions
as the curves in the top panels, i.e Pyx . No significant dependence on the position is observed. Bottom right panel: raw counts from the FP
frame normalized by the corresponding trace profiles, i.e., the quantity Cyx /Pyx . The gray dotted curve shows the inverse-variance weighting Wyx var
win
(arbitrary scale). The solid gray curve represents the full weighting function including the window function Wyx , which limits the extraction to
±5 pixels (vertical dotted lines).
More detailed insights can be gained by inspecting cuts one universal trace profile that is independent of the SED of
through the trace in detector X direction (cross-dispersion direc- the source, which is the fundamental assumption of the Horne
tion) and Fig. 4 shows a series of such cuts. The central pro- (1986) optimal extraction scheme, is therefore not given. This
file (shown in the strongest color) coincides with the peak of an demonstrates that the spectral shape of the source and the trace
arbitrary chosen Fabrry-Pérot line. In addition, eight additional profile observed on the detector are coupled and not separable.
profiles are shown, which are offset in detector Y direction The observed behavior leads to undesired effects when
(wavelength direction) by up to ±4 pixels in 1 pixel steps. averaging the flux estimates in cross-dispersion direction. The
The top-left panel of Fig. 4 illustrates these cuts in (bias and bottom-right panel of Fig. 4 shows the bias and background
scattered-light corrected) raw counts. Obviously, the cut through subtracted counts normalized by the trace profile, which corre-
the center of the FP line shows higher flux than the ones through sponds to Cyx /Pyx as given in Eq. (3), for nine detector columns.
the wings of the line. In addition, it is clear that the trace pro- All pixels of a given cut should represent the same unbiased esti-
files are not Gaussian. Instead, they are flatter at the center and mate of the total received flux at that detector Y position. Thus,
then drop off faster than a Gaussian at the flanks. However, this the plotted lines should be flat with just increasing noise away
behavior is not identical for all nine profiles. from the center of the trace. However, one finds the already men-
To highlight this difference in shapes, the top-right panel of tioned double-horn profile with lower flux in the center of the
Fig. 4 shows the same cuts normalized to unity integrated flux. trace compared to about ±5 pixels above or below. This is at
It becomes clear that the flattening of the profile at the center is least true for the cuts that go through the center of the FP line.
more pronounced for the central profiles while the cuts through Cuts through the wings of the line show the opposite shape, with
the wings of the line are actually more Gaussian. This clearly the regions about ±6 pixels away from the center showing less
indicates that the trace profile changes across the FP line. flux than the center itself. This will certainly impact the accuracy
However, this is not related to a change of the trace pro- of the extracted spectra and possibly lead to biased estimates.
file with detector position but instead intrinsic to the emission In particular, the chosen width of the extraction window now
line. To demonstrate this, the bottom-left panel of Fig. 4 shows directly influences the extracted flux and the resulting spectral
cuts at the identical positions but extracted from the master flat- line shape. It has to be stressed that the extraction process works
field. Clearly, they are all identical, showing that the trace pro- perfectly fine for broadband spectra, but shows the illustrated
file for a white-light source does not vary, while for the FP line systematics for spectra that are not flat. This will unavoidably
the trace profiles changes. This discrepancy is purely related have an impact on how spectral features will be recovered and
to the spectral shape of the source, most importantly showing therefore on the wavelength calibration and the measurement of
narrow emission line vs. broadband emission. The existence of quasar absorption lines.
A144, page 7 of 26A&A 646, A144 (2021)
If now trace profiles determined from a flatfield frame are
5 used to normalize the flux corresponding to an isolated emission
Detector X
15
0
line, for example from a FP or LFC spectrum, the resulting nor-
malized flux will not show the desired flat shape. Instead, it will
10
−5 resemble a double-horn profile for the position right at the center
of the line and a bell shape for its wings. This is illustrated in the
5 right panel of Fig. 5, which shows the profiles from the central
5
Detector X
Detector X
row divided by the corresponding trace profiles from the bottom
0 0 row, which corresponds to the quantity Cyx /Pyx given Eq. (1).
−5
Although rather simple, this toy model shown in Fig. 5 repro-
−5 duces at least qualitatively the effects found on real Espresso
data to a remarkably high degree (compare to Fig. 4). It demon-
5 −10 strates that for roundish fibers, a mismatch between trace pro-
Detector X
0
files naturally and unavoidably occurs as soon as the spectral
−15 shapes deviate (spectrally flat broadband source vs. individual
−5 emission line) and also illustrates the fundamental limitation
−5 0 5 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80
of the adopted extraction scheme based on Horne (1986) and
Detector Y Flux Extracted Flux Zechmeister et al. (2014). The use of rectangular fibers might
(similar to a slit) reduce this particular problem since in this case
Fig. 5. Simple model illustrating the mismatch of trace profiles between the trace profile should depend far less on the position at which it
FP and flatfield spectra. The left column shows the flux distribution intersects the pseudo-slit image. However, the only proper solu-
on the detector for a monochromatic source fed to an idealized fiber- tion would be to model the flux distribution on the detector as
spectrograph (top), the same monochromatic source affected by some superposition of individual 2D point-spread functions weighted
instrumental blurring (center), and a spectrally-flat broadband source
by the spectral energy distribution of the source.
(bottom). The central column shows cuts through these flux distribu-
tions in cross-dispersion direction at different locations. The positions of An algorithm that actually does this is Spectro-Perfectionism
the cuts are indicated in the left panels by vertical cyan lines. Decreasing described by Bolton & Schlegel (2010). However, this approach
color saturation corresponds to increasing offset from the center. The is conceptually and computationally extremely challenging and
right panel displays the cross-sections from the central row divided by was only recently adopted by the Dark Energy Spectroscopic
the profiles shown in the bottom row of panels. This corresponds to the Instrument (DESI) consortium. In addition, such an approach
normalization of FP spectra by trace profiles extracted from a flatfield would, despite the tremendous requirements on computational
spectrum. Clearly, there is a discrepancy between the trace profiles and resources, also pose very significant demands on the calibra-
the normalized flux does not exhibit the desired flat distribution along tion procedure, which has to deliver a model of the 2D point-
cross-dispersion direction. spread function for every wavelength of interest and therefore
every location on the detector. It is unlikely that this can be done
These rather complicated properties of the trace profile can using the Fabry-Pérot interferometer, since its lines are resolved
actually be understood considering the design of the spectro- by Espresso. Also, the number of unblended and sufficiently
graph. We therefore show in Fig. 5 a toy model, illustrating bright ThAr lines produced by the hollow-cathode lamps used
the basic principles that lead to a mismatch in the trace profile for wavelength calibration is extremely low and most-probably
between flatfield and FP spectra. too sparse for a proper characterization of the PSF. In addi-
For an idealized spectrograph and a monochromatic light tion, also the ThAr lines are marginally resolved. The laser
source, the flux distribution on the detector is a direct image of frequency comb would in principle be suitable to characterize
the (pseudo-)slit, in this case of the optical fiber feeding the spec- the Espresso line-spread function since it delivers a plethora
trograph. This is shown in the top-left panel of Fig. 5 (assum- of extremely narrow ('100 kHz) lines. However, it covers only
ing for simplicity a circular fiber instead of the octagonal ones 57% of the Espresso wavelength range and the individual lines
utilized in Espresso). A cross-section in cross-dispersion direc- might not be sufficiently separated (∆νLFC = 18 GHz) to fully
tion through the flux distribution would resemble a top-hat pro- disentangle them. For the time being, one therefore has to pro-
file. If the cross-section goes right through the center of this ceed with the classical extraction scheme based on Horne (1986)
observed emission line, the width of the top-hat corresponds to and Zechmeister et al. (2014) described above but be aware that
the full diameter of the fiber, while taking a cut a few pixels off- it is nonoptimal when it comes to details.
set from the flux center would result in a narrower top-hat. A
series of such cuts is displayed in the central-top panel of Fig. 5.
The introduction of diffraction and instrumental aberrations 4. Wavelength calibration
(here approximated by simple Gaussian blurring) will soften the After extraction, a full Espresso exposure is represented as 340
flux distribution on the detector. Still, the cross-section through 1D spectra (Order 161–117 from blue arm, Order 117–78 from
the center of the emission line is wider and fatter compared to red arm, ×2 fibers, ×2 slices), in the terminology of the DRS
the off-center profiles, which are closer in shape to a Gaussian. called the S2D format14 . Each individual spectrum has a length
This is illustrated in the central and central-left panel of Fig. 5. of 9232 pixels in detector Y direction and the basic assumption
For a broadband source, the flux distribution on the detec- of the extraction procedure is that these can be directly mapped
tor can be seen as a superposition of many point-spread func- to wavelengths.
tions corresponding to the individual wavelengths (indicated To do so, the Espresso calibration plan includes three types
with multiple red circles). In case of a spectrally flat source, all of wavelength calibration frames:
cross-sections will have an identical shape, independent of their
position. This case is shown in the bottom row of Fig. 5 and cor- 14
Despite the misleading description, S2D does not describe 2D spectra
responds to a spectral flatfield frame. but a collection of single-order, unmerged one-dimensional spectra.
A144, page 8 of 26T. M. Schmidt et al.: Towards an accurate Espresso wavelength calibration
Firstly, exposures of a Thorium-Argon (ThAr) hollow cath- LC − Arm : Blue, Order : 127, Fiber : A, Slice : a
ode lamp provide absolute wavelength information by the means 0.030
of Th i emission lines. These quantum-mechanical transitions
0.025
have been accurately measured to '5 m s−1 in laboratory experi-
ments using Fourier-transform spectroscopy (e.g., Redman et al. 0.020
Flux
2014). However, many thorium lines are blended, contaminated
by argon lines or simply not strong enough. The default line list 0.015
therefore contains only 432 unique Th i lines. In consequence, 0.010
some Espresso orders are covered by only two thorium cali-
bration lines. This is by itself clearly not enough to derive an 0.005
accurate and precise wavelength solution.
0.000
Secondly, the Fabry-Pérot interferometer (FP) therefore 4750 4800 4850 4900 4950 5000 5050 5100
complements the information from the ThAr frames. It pro- Detector Y Coordinate
duces a dense series of rather narrow, nearly equally-spaced and Fig. 6. Small part of a laser frequency comb spectrum, showing the
equally-bright (apparent) emission lines across the full wave- dense forest of equally-spaced narrow emission lines. The flux is given
length range (Wildi et al. 2010, 2011, 2012). It was designed relative to the flatfield. The LFC spectrum exhibits a significant back-
to produce lines with a separation of ≈1.96 × 1010 Hz, which ground light contribution (green) and modulation of the line intensities
corresponds to 0.1 Å–0.4 Å. The lines are therefore marginally (red), which are both modeled as part of the analysis with cubic splines.
resolved and noticeably broader than the instrumental profile.
The wavelengths are purely defined by the effective optical
length of the Fabry-Pérot cavity ('15.210 mm). However this wavelength range16 . A concept based on two LFCs (3800–
might change due to variations in temperature or pressure. 5200 Å and 5200–7600 Å) to cover the full spectral range of
Without stabilization to any reference, the FP provides no Espresso, as initially envisioned (Mégevand et al. 2014), is
absolute wavelength information. Instead, it has to be character- unfortunately not realized. In addition, the LFC is still rather
ized by comparison to ThAr frames. Despite this, the large num- unreliable with only '25% availability since the start of regu-
ber of densely-spaced FP lines (about 300 per order) allows a lar observations in October 2018.
very precise wavelength calibration on small scales, completely These wavelength calibrations, as well as flatfield and bias
impossible with arc lamps alone. Therefore, ThAr and FP pro- frames, are taken daily, usually in the morning, as part of the
vide highly complementary information and combining both standard Espresso calibration scheme. We make sure that all
allows to derive a precise and accurate wavelength solution over exposures processed together are from the same calibration ses-
the full Espresso wavelength range15 , which is in the following sion and therefore taken within less than 2.5 hours. Instrumental
denoted as ThAr/FP solution. drifts are therefore negligible.
Thirdly, a completely independent means of calibration is
provided by the laser frequency comb (LFC), a passively mode- 4.1. Line fitting
locked laser that emits a train of femtosecond pulses producing
To process the wavelength calibration frames, the flux is
a set of very sharp emission lines (Wilken et al. 2010a, 2012;
extracted in the way described in Sect. 3 and in addition
Probst et al. 2014, 2016). The frequencies of the individual lines
de-blazed. This is essential to avoid a biasing of the determined
νk follow exactly the relation νk = ν0 + k × νFSR . The off-
line positions. The blaze function is determined by optimal
set frequency ν0 and separation of the lines νFSR are actively
extraction from the master flatfield, identical to the way all other
controlled and compared to a local (radio) frequency standard,
spectra are extracted. It therefore contains all sensitivity varia-
which itself is stabilized against an atomic clock or GPS. There-
tions in detector Y direction (wavelength direction), in particular
fore, the accuracy of the fundamental time standard is trans-
column-to-column and large-scale variations due to the blaze or
ferred into the optical regime, in principle providing absolute
transmission properties of the spectrograph, while the trace pro-
calibration with and accuracy at the 10−12 level. However, this
file as outlined in Sect. 3.1 captures sensitivity variations across
extreme accuracy of the laser frequency comb itself does not
the trace, which is in detector X direction. All extracted spec-
translate one-to-one into the accuracy of the final wavelength
tra are normalized by the blaze function determined in this way.
solution of the spectrograph and the details of this process are
Thus, the fluxes stated in the following are relative to the flux
outlined in the following sections. The LFC is operated with a
of the flatfield light source17 , which provides a rather featureless
mode-spacing of νFSR = 18.0 GHz and an offset frequency of
and flat spectrum.
ν0 = 7.35 GHz. The spacing of the LFC lines on the detec-
After extraction and de-blazing, the individual emission lines
tor is therefore very similar to the ones of the FP. In addition,
are fitted. For the ThAr spectrum, this is done based on a line list
the LFC lines have an extremely narrow width of '100 kHz,
with good initial guess positions. Lines are fitted with Gaussian
corresponding to less than ' 10 1000 of the spectrographs reso-
functions plus a constant offset within a window of ±10 km s−1
lution in 1HR mode. It is therefore the only calibration source
around the initial guess position. The used ThAr line list is iden-
that allows an accurate characterization of the instrumental line-
tical with the one used by the Espresso DRS (Lovis et al., in
spread function and spectral resolution. However, due to funda-
prep.) and the laboratory wavelengths and their uncertainties are
mental technical challenges and design choices, the Espresso
taken from Redman et al. (2014).
LFC covers only ≈57% of the wavelength range, with signifi-
Fitting of FP and LFC lines is done in an identical way. Start-
cant flux only from Order 132 (4635 Å) to Order 85 (7200 Å), ing from the center of an order, line peaks are identified and
and substantial uncovered regions at the blue and red end of the fitted, again with a Gaussian plus constant background model.
15 16
This of course requires that there is no significant instrumental drift We stress that the LFC flux levels are not stable and the usable wave-
between ThAr and FP exposures. However, this is no issue within the length range can change substantially from epoch to epoch.
17
context of this study. An Energetiq laser-driven light source (LDLS) EQ-99X.
A144, page 9 of 26You can also read
