# Spectral Line Depth Variability in Radial Velocity Spectra

Alexander Wise Department of Astronomy and Astrophysics, The Pennsylvania
State University, University Park, PA 16802, USA  Center for Exoplanet and
Habitable Worlds, The Pennsylvania State University, University Park, PA
16802, USA  Peter Plavchan Department of Physics and Astronomy, George Mason
University, Fairfax, VA 22030, USA  Xavier Dumusque Astronomy Department,
University of Geneva, 51 Chemin des Maillettes, 1290 Versoix, Switzerland
Heather Cegla Physics Department, University of Warwick, Coventry CV4 7AL,
United Kingdom  Astronomy Department, University of Geneva, 51 Chemin des
Maillettes, 1290 Versoix, Switzerland  Duncan Wright University of Southern
Queensland, Centre for Astrophysics, Australia

###### Abstract

Stellar active regions, including spots and faculae, can create radial
velocity (RV) signals that interfere with the detection and mass measurements
of low mass exoplanets. In doing so, these active regions affect each spectral
line differently, but the origin of these differences is not fully understood.
Here we explore how spectral line variability correlated with S-index (Ca H &
K emission) is related to the atomic properties of each spectral line. Next we
develop a simple analytic stellar atmosphere model that can account for the
largest sources of line variability with S-index. Then we apply this model to
HARPS spectra of $\alpha$ Cen B to explain Fe I line depth changes in terms of
a disk-averaged temperature difference between active and quiet regions on the
visible hemisphere of the star. This work helps establish a physical basis for
understanding how stellar activity manifests differently in each spectral
line, and may help future work mitigating the impact of stellar activity on
exoplanet RV surveys.

line: profiles — planets and satellites: detection — stars: activity — stars:
variables: general — starspots

##  1 Introduction

The radial velocity (RV) method has been a prolific method of confirming and
discovering exoplanets in the past two decades (see e.g. Wright (2017) for a
review). State-of-the-art Extreme Precision Radial Velocity (EPRV) instruments
have instrumental precision of 20-30 cm/s (Pepe et al., 2013; Jurgenson et
al., 2016; Blackman et al., 2020; Suárez Mascareño et al., 2020), but stellar
activity generates RV signals with amplitude 0.5-5 m/s in typical exoplanet
survey stars (e.g. Dumusque et al., 2011). Stellar activity is defined here as
any spatial or temporal variability in brightness or velocity on the visible
stellar disk that can manifest as a detectable RV signal. Some types of
stellar activity occur on timescales that make averaging them down with many
observations unfeasible (see e.g. Plavchan et al. (2020) for a summary).
Granulation and super-granulation vary on timescales of minutes to hours (De
Rosa & Toomre, 2004; Brandt & Getling, 2008; Miklos et al., 2020) and magnetic
activity such as spots, faculae and plage vary on timescales of hours to years
(Baliunas et al., 1995; Saar & Donahue, 1997; Haywood et al., 2020). To
improve RV precision beyond the current limits set by stellar activity, the
imprint of stellar activity on RVs and their underlying spectra must be
understood.

Whole-spectrum cross-correlation function (CCF) line shape indicators have
been used to diagnose and characterize stellar activity (e.g. de Beurs et al.,
2020; Collier Cameron et al., 2021). However, several studies have looked at
individual spectral line variability to find out how stellar activity affects
each line differently. Thompson et al. (2017) have found for $\alpha$ Cen B,
and now the Sun (Thompson et al., 2020), that the depth of some spectral lines
were significantly correlated with emission in the cores of the Ca II H & K
lines. Similarly, Wise et al. (2018) (hereafter W18) used the same data from
$\alpha$ Cen B to measure correlation coefficients between spectral line
shapes and the Ca II H & K lines. Dumusque (2018) found that the RVs of some
spectral lines were much more affected by stellar activity than others, and by
curating the spectral lines on which the stellar RV is calculated, it is
possible to mitigate stellar activity by a factor of two. Finally, Cretignier
et al. (2020) pushed further the analysis performed in Dumusque (2018) and
found that the lines the most affected by stellar activity were the shallow
ones, formed deep inside the stellar atmosphere, where convection is strong
and therefore is strongly affected by the magnetic field responsible for
stellar activity. Line-by-line analysis seems to be a promising tool to help
us understand and model stellar activity (e.g. Cretignier et al., 2021).

Here we push further the results of W18 by developing an astrophysical toy
model of their observed spectral line variability. W18 analyze High Accuracy
Radial velocity Planet Searcher (HARPS) spectra of $\alpha$ Cen B during a
period of high magnetic activity in 2010 identified in Dumusque et al. (2012).
They quantify activity-induced line variability as the Kendall’s Tau
correlation between the Mt. Wilson S-index (Wilson, 1978) and each line’s
depth, width, or center of mass. The S-index measures Ca H & K emission and is
sensitive to magnetic activity. W18 found that generally line depth variations
had the most significant correlations with S-index. Our primary aim is to
develop an astrophysical model to explain these most significant spectral line
variations with activity. We also investigate relationships between
correlation coefficients measured in W18 and several atomic properties of
spectral lines.

The paper is organized as follows. In Section 2, we investigate relationships
between the atomic properties of spectral lines and correlations linking
spectral line measurements to S-index. In Section 3, we develop an analytic
model to explain the interaction between line core flux vs. S-index
correlation coefficients and excitation energy. Finally, in Section 4, we
summarize our findings and discuss future work.

##  2 Correlations with Atomic Properties

Here we search for correlations between activity-related line variations and
atomic properties of spectral lines. These properties can tell us how lines
change in response to variations in the solar surface, such as temperature or
magnetic field strength changes. We use correlation coefficients between
S-index and line depth, width, and center of mass. First, we summarize how
these correlation coefficients were calculated in W18. Then we compare these
correlation coefficients to spectral line and atomic properties extracted from
the Vienna Atomic Line Database (VALD) (Kupka et al., 1999; Piskunov et al.,
1995).

W18 used a data set of HARPS spectra of $\alpha$ Cen B from March-June 2010.
In W18, after the order-by-order spectra were continuum normalized by fitting
a polynomial to a set of local maxima, an automated pipeline was used to
identify wavelength intervals that were either stellar atomic spectral lines
or blends by first selecting all pixels in the normalized spectrum below a
value of 0.8, and then grouping selected pixels within
$\Delta\lambda/\lambda=10^{-5}$ (i.e. 3 km/s) of their nearest selected
neighbor into wavelength intervals. Then the line core flux (1 - depth), half-
depth range (line width at half-depth flux), and center of mass (area-weighted
center below half-depth) were measured for each wavelength interval, e.g.
Figure 1. Intervals were rejected if their half-depth flux was greater than
the flux of the pixels at the interval edges, effectively only keeping
intervals with depth $\gtrsim$ 40%). Kendall’s Tau correlation coefficients
111The tau-b version of Kendall’s Tau was calculated in python using the scipy
package. were calculated between each of these three measurements for each
line and the S-index.

Figure 1: Graphic showing how line properties were measured in W18. The
continuum has a normalized flux of 1, and this line has a depth of 0.94. Half-
depth flux is a constant value taken from a normalized master spectrum
computed as the median of all individual normalized spectra. There are three
time-varying measurements. Line core flux is the minimum of a cubic spline
interpolation on continuum normalized order-by-order spectra. Half-depth range
is the width of the line at half-depth flux. Center of mass is the weighted
barycenter in wavelength of the diagonally hatched area.

Our analysis is restricted to the deepest spectral line within each wavelength
interval as follows. For each wavelength interval, spectral lines and their
expected depths are obtained using a VALD “extract stellar” query using
stellar parameters for $\alpha$ Cen B222The “extract stellar” star parameters
we used were: microturbulence: 1.1 km s-1, $T_{\rm eff}$: 5200 K, $\log{g}$:
4.5 (g in cgs units), and chemical composition: “Fe: -4.34”. The last
parameter is VALD’s default stellar iron abundance plus 0.2 dex (Porto de
Mello et al., 2008)., with a threshold line depth $>$ 0.1. Out of all of the
VALD lines within $2\times 10^{-5}\times\lambda$ (i.e. 6 km/s), where
$\lambda$ is the wavelength of minimum flux within each wavelength interval,
the line of maximum VALD depth is treated as the only line within that
interval for the extraction of atomic properties from VALD. We also remove the
first 30 out of 72 HARPS orders (approx. 3800 to 4650 Å) from this analysis
due to significant line blending in those orders, and remove lines with
excitation energy greater than 8 eV to exclude the Balmer series. As a result
we consider 636 spectral lines in this section, with wavelengths ranging
$\sim$4650 to 6900 Å.

The atomic and spectral properties of each line are retrieved from the VALD
query. These properties are:

  1. 1.

Wavelength in air (Å).

  2. 2.

Line depth: Ratio between line absorption and continuum absorption at the
line’s wavelength.

  3. 3.

Excitation energy: Energy required to reach the lower electronic state of the
transition from the ground state (eV).

  4. 4.

Oscillator strength: log($gf$) where f is the absorption/emission probability
and g is the statistical weight of the lower energy level.

  5. 5.

Landé factor: Factor determining the magnitude of energy level splitting in a
weak magnetic field (no units).

  6. 6.

Radiation damping coefficient: The rate of decay of the transition, expressed
as the logarithm of the radiation damping constant in (4$\pi$s)-1.

  7. 7.

Stark damping coefficient: Also known as the pressure broadening constant,
expressed as the logarithm of the Stark damping constant in (4$\pi$s
$N_{\mathrm{e}}$)-1 at 10,000 K (where $N_{\mathrm{e}}$ is the number density
of electrons)

  8. 8.

Van der Waals damping coefficient: Describes the strength of interaction
between neutral atoms, expressed as the logarithm of the Van der Waals damping
constant in (4$\pi$s $N_{\mathrm{H}}$)-1 at 10,000 K (where $N_{\mathrm{H}}$
is the number density of neutral Hydrogen)

In Figure 2, we plot the Kendall’s Tau correlation coefficients as in W18 on
the vertical axes, and each of these eight properties on the horizontal axes.
These taus on the vertical axis represent how activity sensitive each line’s
depth (line core flux), width (half-depth range), and RV (flux deficit-
weighted center) are. Therefore, these plots show how each line’s activity
sensitivity depends on each on the eight properties above.

Now we look for relationships between these correlation coefficients and the
spectral line and atomic properties of each wavelength interval. We quantify
the strength of each relationship using Kendall’s Tau (tau-b version, Stuart,
A., 1953) as the underlying data do not appear to be normally distributed.
Hence we are calculating a new tau, printed above each plot in Figure 2,
between each vertical-axis tau and each horizontal-axis line property. Figure
2g has the largest magnitude of these new correlation coefficients, $|\tau|$ =
0.41, and by eye appears moderately significant. In the next section, we
explore an astrophysical motivation for this trend using a toy model for line
depth changes derived from principles of stellar atmospheres.

Figure 2: Plots visualizing the relationships between each line measurement in
W18 and each line property we extract from VALD. These plots are provided to
show why we choose the trend in (g) for deeper analysis. Above each plot is
listed a new tau-b value for the correlation between the y-axis tau and the
x-axis line property.

##  3 Modeling Line Depth Variations

The strongest correlation in the previous section is between electronic
excitation energy and the Kendall’s Tau correlation coefficients between line
core flux and S-index, as seen in Figure 2g. Here we explore that relationship
in more detail. First we describe the analytic stellar atmosphere model that
we adopt to explain Figure 2g. Then we apply this model, and discuss what we
can learn about small variations in the disk-averaged temperature of $\alpha$
Cen B based on this model and these observations.

###  3.1 Stellar Atmosphere Model

Here we model changes in line depth with excitation energy as arising from
changes in disk-averaged temperature, considering changes in absorption and in
continuum thermal emission for each line. For the absorption, we follow the
analytic model developed in Gray (2008), equation 13.21. We calculate the
ratio, $R$, of line absorption, $l_{\nu}$, to continuous absorption,
$\kappa_{\nu}$:

$R=\frac{l_{\nu}}{\kappa_{\nu}}.$ (1)

The line absorption $l_{\nu}$ is proportional to the number of absorbers, i.e.

$l_{\nu}=\mathrm{constant\ }e^{-\epsilon/kT}$ (2)

where $\epsilon$ is the excitation energy of the lower state of the line’s
transition, $k$ is boltzmann’s constant, and $T$ is temperature. For the
visible region of FGK stars, we can model $\kappa_{\nu}$ as the negative
hydrogen ion’s bound-free absorption, i.e.

$\kappa_{\nu}=\mathrm{constant\ }T^{-5/2}\ P_{\mathrm{e}}\
e^{0.75\mathrm{eV}/kT}$ (3)

$T$ is temperature, $P_{\mathrm{e}}$ is the electronic pressure, and 0.75 eV
is the binding energy of the negative hydrogen ion. We approximate the
electron pressure as

$P_{\mathrm{e}}\approx\mathrm{constant\ }e^{\Omega T}$ (4)

where $\Omega=0.0015$ for FGK dwarfs at an optical depth $\tau=1$ (Gray,
2008). Combining equations 1 – 4 and taking a natural log gives

$\mathrm{ln}R=\mathrm{constant}+\frac{5}{2}\mathrm{ln}T-\frac{\epsilon+0.75\mathrm{eV}}{kT}-\Omega
T.$ (5)

This gives us the equation for the change in line depth with temperature due
to absorption:

$\frac{1}{R}\frac{\mathrm{d}R}{\mathrm{d}T}=\frac{2.5}{T}+\frac{\epsilon+0.75\mathrm{eV}}{kT^{2}}-\Omega.$
(6)

Next we model the continuum emission using the Planck function

$B_{\nu}=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/kT}-1}.$ (7)

The fractional rate of change in emission with temperature is

$\frac{1}{B_{\nu}}\frac{\mathrm{d}B_{\nu}}{\mathrm{d}T}=\frac{h\nu}{kT^{2}}\frac{1}{1-e^{-h\nu/kT}}.$
(8)

Putting together the equations for continuum thermal emission and line
absorption, we obtain the model of how a line core’s normalized flux, $F$,
changes with temperature:

$\frac{\mathrm{d}F}{\mathrm{d}T}=\frac{1}{B_{\nu}}\frac{\mathrm{d}B_{\nu}}{\mathrm{d}T}-\frac{\mathrm{d}R}{\mathrm{d}T}=\frac{h\nu}{kT^{2}}\frac{1}{1-e^{-h\nu/kT}}-R\left[\frac{2.5}{T}+\frac{\epsilon+0.75\mathrm{eV}}{kT^{2}}-\Omega\right].$
(9)

In the middle part of this equation, the term $dR/dT$ does not have a $1/R$
factor (as the $B_{\nu}$ term does) since it is already normalized by the
absorption of the continuum in Eq. 1. For the right side of the equation, Gray
(2008) suggest using a temperature about 15% below the effective temperature
as characteristic of the line forming region, so starting with
$T_{\mathrm{eff}}=5230$ K for $\alpha$ Cen B (Brandenburg et al., 2017) gives
us $T=4445.5$ K. Note that we also tried using $T=5230$ in the model, and our
resulting $\Delta T$ in Section 3.2 below changed from 3.3 K to 3.6 K, a $<$
10% difference in our result. In applying our model, we set the VALD values
for line depth equal to $R$ in Eq. 9, as our formulation of $R$ is a good
approximation for line depth for “weak lines of a neutral species with the
element mostly neutral” as shown in Gray (2008), which are the types of lines
we consider below. Next we use this model, along with the results of W18, to
learn about disk-averaged temperature variations in $\alpha$ Cen B.

###  3.2 Fitting the Data

Here we apply the above line depth model to the line depth measurements from
W18. To simplify the application of the model, we first apply the following
two filters to our list of 636 spectral used in Section 2. First, we remove
all blends where the range in excitation energies of spectral lines in the
blend is greater than 1 eV. This eliminates “line mislabeling” cases where a
shallow, low-excitation line dominates the line depth variations over a deeper
line in a blend. Then we remove all spectral lines other than Fe I lines, as
Fe I lines account for a large fraction of lines in the visible spectrum and
are primarily formed in the stellar photosphere. After these two filters, 236
iron lines remain. We remove eight additional lines that are clearly
contaminated by tellurics, as described in the next section. As a result, 228
iron lines are used the following analysis.

Figure 3 shows the result of our model calculations of d$F$/d$T$ for all of
the iron lines in our sample. We see increased temperature sensitivity at
lower excitation potential as in previous models and solar observations (e.g.
Rutten & Zwaan, 1983; Elste, 1986; Takeda & UeNo, 2017). This model tells us
how line core flux changes with temperature, while W18 explored how line core
flux changes with S-index, $S$ (e.g. as in Brandt & Solanki, 1990). To fit the
model to the line core flux measurements for $\alpha$ Cen B, we consider the
equation

$\frac{\mathrm{d}F}{\mathrm{d}S}=\frac{\mathrm{d}T}{\mathrm{d}S}\frac{\mathrm{d}F}{\mathrm{d}T}.$
(10)

We calculate d$F$/d$S$ using the line core flux and S-index measurements in
W18. Looking at the correlation plots in W18, an example of which is shown in
Figure 4, it seems a linear fit is a good approximation for the relationship
between S-index and line core flux. Hence we use a weighted least squares
regression to find the slope of the line of best-fit for each Fe line in our
sample. The weights are the signal-to-noise ratios (SNRs) per pixel,
calculated by taking the average over the middle 200 pixels of order 35 in
each spectrum. We plot these slopes on the vertical axis of Figure 5-left.

Figure 3: The change in line core flux with temperature (d$F$/d$T$) for the
model is plotted versus the lower state excitation energy. The two iron lines
near the coordinates (4.8,0.0012) are distinct because of their small VALD
depths ($\sim$0.15).  Figure 4: An example correlation plot from W18. This
plot demonstrates that a linear fit is a good approximation for the
relationship between S-index and line core flux.

The only term we have not calculated in Eq. 10 is d$T$/d$S$, the change in
temperature with S-index. We assume this is a constant within the range of
S-index values for this data set, as this is the simplest approximation we can
make and any consideration of a more complicated relationship is beyond the
scope of this work. One consequence of this assumption is that Figures 3 and
5-right appear identical (except for different y-axis scales). d$T$/d$S$ is
calculated by dividing the slopes of best-fit lines to Figures 3 and 5-left.

Lastly, we calculate the temperature change between the more “active”
hemisphere (rotational phase with highest S-index) and “quiet” hemisphere
(lowest S-index) of the star using

$\Delta T=\frac{\mathrm{d}T}{\mathrm{d}S}\Delta S$ (11)

where $\Delta S$ is the full range of S-index values in our data. To aid in
the interpretation of this $\Delta T$, we considered modifying the model for
d$F$/d$T$ to include a constant multiplying the whole equation, $A$, which we
interpret as the fractional active area on the disk. However, $A$ was found to
be degenerate with the temperature according to

$A\ \Delta T\approx\mathrm{constant}.$ (12)

Thus, our model cannot distinguish between a 300 K hotter active region
covering 1% of the disk and a 3 K hotter active region covering 100% of the
disk. Consequently, we interpret the temperature $T$ as the disk-averaged
temperature.

Figure 5 shows a side-by-side comparison of our measured values of d$F$/d$S$
with our model for d$F$/d$S$ calculated using Eqs. 9 and 10. Note that because
we calculated d$T$/d$S$ by dividing the slopes of best-fit lines to Figures 3
and 5-left, the slope of the right-hand panel of Figure 5 has been
artificially matched to the slope of the left-hand panel. However, this
process does not align the vertical intercepts of the two plots (value of
d$F$/d$S$ when lower state energy = 0). Therefore, the qualitative agreement
of the intercepts between the two panels is encouraging. We also color-code
Figure 5 based on line depth to show the depth-dependence in the model. Figure
6 shows the correlation plot for the model and data. In the next section, we
discuss possible reasons for the differences between the model and data.

Figure 5: Comparison of our measured values of d$F$/d$S$ on the vertical axis
(left panel) to our best-fit model for d$F$/d$S$ (right panel), both as a
function of lower state excitation energy. The color scale shows line depth
and is truncated at 0.5 to improve visualization for the majority of data
points. The two model points near the coordinates (4.8,0.07) are distinct
because of their very small VALD line depths ($\sim$0.15). Note, that the
vertical axis on the right panel is not the same as shown in Figure 3, but is
scaled by a constant d$T$/d$S$.  Figure 6: Correlation plot of our best-fit
model vs. measured values of d$F$/d$S$.

###  3.3 Discussion

We have found a simple stellar atmospheric model of line formation helps
explain why some lines have depth variations that are more correlated with
S-index than others. As the values of d$F$/d$S$ calculated from the data have
a much larger scatter within a small bin of excitation energy than the model
values, as seen in Figure 5, here we discuss possible sources of error.
Sources of error include blending of spectral lines, which can affect how each
line’s core depth changes with excitation energy if the line is blended with a
line of differing excitation energy, or of a different species. While we
attempt to mitigate the effects of line blending by only choosing blends where
the deepest line is Fe I and the range in excitation energy is $<$ 1 eV, we
make no attempt model these effects.

Another source of error in this analysis is telluric contamination. This is
not an issue in W18 as they find no evidence of telluric contamination for
correlation coefficients $\tau>0.5$. However, the full range of $\tau$ is
considered here, so an attempt is made to remove outliers that are most
clearly caused by telluric lines.

To flag lines with clear evidence for telluric contamination, first we look at
time-series plots of line core flux to identify candidates for telluric
contamination, e.g. Figure 7-Left. To confirm these are tellurics, we then
created and watched videos of each candidate spectral line, where the telluric
can be seen gradually moving into and/or out of the line due to the annual
variation of the barycentric Earth RV. Thirty iron lines that have the largest
y-coordinate difference between the left- and right-hand sides of Figure 5 are
considered for this analysis. Eight out of these thirty lines are found to be
contaminated by tellurics, and all eight come from the eleven largest
differences, so we do not consider additional lines after these thirty. In the
future, an automated approach could be used that considers all lines and
measures the velocity range over which the line core flux variations occur, as
shown in Figure 7-Right. Comparing this velocity range to the typical telluric
line’s velocity width could help discriminate between telluric contamination
and other signals.

Figure 7: Plots showing the effect of a telluric line on the flux measured in
the core of a Fe I line. Left: Line core flux plotted against time. As the
telluric line passes through the Fe I line’s core over time, the Fe 1 line
appears deeper between JD-2455000 of 320 and 350. Right: Line core flux
plotted against barycentric RV. The telluric line and stellar line only “line
up” for a small window of RVs, and the full width at half maximum (FWHM) of
that window matches the telluric line’s $\sim$4 km/s FWHM.

If we had not removed these eight telluric-contaminated lines, several of them
would correspond to data points outside of the vertical axes of Figure 5-left.
Now, there are no data points outside of these axes. Therefore we suggest that
some of the remaining scatter is due to telluric lines that are too weak to be
identified by the visual analysis described above.

The following are also suspected to be causes of error in our d$F$/d$S$
measurements, though we were unable to find convincing evidence of these
effects. Iron lines at bluer wavelengths may have weaker responses to
temperature than predicted by the model due to an incorrect line formation
temperature (we estimated this as 85% of $T_{\mathrm{eff}}$ for all lines).
This issue may have been mitigated by discarding the first 30 out of 72 HARPS
orders. Also, poor continuum normalizations in line-blanketed orders or higher
photon noise at the bluer orders may contribute to the scatter in our
measurements of d$F$/d$S$. In the next section we summarize the results of
this paper and discuss future avenues of research.

##  4 Conclusions and Future Work

We searched for correlations between the atomic properties of spectral lines
and the correlation coefficients between S-index and line depth, width and
barycenter. W18 derived these S-index correlation coefficients using HARPS
spectra of $\alpha$ Cen B during a time of high, sinusoidally varying
activity. We identify a particularly interesting correlation between the line
core flux correlation coefficients and the lower state excitation energy, a
correlation not previously identified. This correlation demonstrates that (Fe
I) lines with smaller excitation energy have more activity-sensitive depths.

We then developed a toy model of the expected line core flux changes with
temperature that qualitatively reproduces the observations. This model can be
used to estimate the temperature difference in magnetically active regions.
The line core flux changes in this particular data set could be explained by
the disk-averaged temperature being 3.3 K hotter when the star shows maximum
activity, compared to minimum activity. In Dumusque (2014), the author shows
that during the time of the data analyzed here, a facula of
2.12${}^{+0.88}_{-0.56}$ % of the visible hemisphere is covering part of the
stellar surface. Linking the 3.3 K average increase in temperature and the
2.12 % size of the active region, we can estimate an active region temperature
contrast of 150 K hotter than the quiet stellar surface using the Stephan-
Boltzmann law333$T_{\mathrm{faculae}}$ is calculated from the Stephen-
Boltzmann law by setting the fluxes equal:
$(T_{\sun}+3.3~{}\mathrm{K})^{4}=(1-0.0212)*T_{\sun}^{4}+0.0212*T_{\mathrm{active}}^{4}$
where $T_{\sun}$ = 5778 K This implies the presence of a large or several
small faculae on the surface of $\alpha$ Cen B. The temperature contrast is
within the large expected range (e.g. 34 K to 251 K from disk center to edge
in Meunier et al., 2010).

As we improve our models of how stellar activity affects RV measurements, we
would like to understand the dominant ways in which stellar activity affects
the spectrum on a physical basis. The present study can be seen as a step
toward physically motivated models of how stellar activity affects every pixel
in a spectrum. W18 found that line core flux measurements had the strongest
correlations with S-index, and now we can partially understand those
correlations using a simple analytic model of stellar atmospheric absorption
and emission. As line depth does not affect CCF-based RV measurements, except
in cases of blended lines or template errors, this line core flux correlation
is distinct from traditional measurements of line shape changes such as
bisector span and chromatic index (e.g. Queloz et al., 2001; Zechmeister et
al., 2018). Our analysis provides a “new” way of looking at how the spectra of
stars due to activity are dynamic, and offers an “old” astrophysical
explanation thereof. In the future, we aim to continue to assign physical
explanations to the ways in which stellar activity affects RV spectra,
including the various measurements of line shape changes.

In traditional Doppler RV analysis, the spectrum of a star is taken to be
static in time. However, we have shown that stellar spectra of interest to
planet hunters are measurably dynamic in response to changes in stellar
activity level. Rajpaul et al. (2020) propose a new data-driven method of
measuring RVs that applies this principle by not requiring a static stellar
template. In future work, we plan to take a look at how changing line depths
can induce RV measurement error due to the mask no longer correctly weighting
each line. The path forward requires treating the stellar spectrum as dynamic
rather than static.

This research is based on data products from observations made with ESO
Telescopes at the La Silla Paranal Observatory under programme IDs
60.A-9036(A), 072.C-0488(E), 072.C-0513(B), 072.C-0513(D), 074.C-0012(A),
074.C-0012(B), 076.C-0878(A), 077.C-0530(A), 078.C-0833(A), 079.C-0681(A),
084.C-0229(A), 085.C-0318(A), and 192.C-0852(A). This research has made use of
the services of the ESO Science Archive Facility, and is based on data
obtained from the ESO Science Archive Facility under request numbers 243402,
243835, 300034, 300050. This work has made use of the VALD database, operated
at Uppsala University, the Institute of Astronomy RAS in Moscow, and the
University of Vienna. A. W. acknowledges support from the Delaware Space Grant
College and Fellowship Program, NASA Grant NNX15AI19H. This work was partially
supported by funding from the Center for Exoplanets and Habitable Worlds. The
Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania
State University and the Eberly College of Science. This research was
partially supported by Heising-Simons Foundation Grant #2019-1177. C X. D.
acknowledges support from the Branco-Weiss–Society in Science fellowship. This
project has received funding from the European Research Council (ERC) under
the European Union Horizon 2020 research and innovation program (grant
agreement No. 851555). P.P. acknowledges support from NASA (award
16-APROBES16-0020 and support from the Exoplanet Exploration Program) and the
National Science Foundation (Astronomy and Astrophysics grant 1716202), the
Mount Cuba Astronomical Foundation and George Mason University start-up funds.
H.M.C acknowledges financial support from the National Centre for Competence
in Research (NCCR) PlanetS, supported by the Swiss National Science Foundation
(SNSF), as well as a UK Research and Innovation Future Leaders Fellowship.

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y&=y+ 0.08771158424235 \cdot  \cos(s) -0.056379754715369 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.34681438881798) + \overline{v_{k-1}}\cdot  0.043514921077717 + \overline{v_{k-2}}\cdot  0.12046630595345 + \overline{v_{k-3}}\cdot  (-0.11064217792073) \\&+ \overline{n_k}\cdot  12.409726448748 + \overline{n_{k-1}}\cdot  0.12883850899569 + \overline{n_{k-3}}\cdot  -13.632000414936 \\&+ \overline{t_k}\cdot  (-0.19971464610846) + \overline{t_{k-2}}\cdot  0.046228428472665 + \overline{t_{k-3}}\cdot  0.12515304947418\\
y&=y-0.055589202591183 \cdot  \cos(s) -0.17750376299084 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.5126466445729 + \overline{v_{k-1}}\cdot  0.12860832423111 + \overline{v_{k-2}}\cdot  (-0.21064805610633) + \overline{v_{k-3}}\cdot  (-0.50620018378431) \\&+ \overline{n_k}\cdot  5.236695403519 + \overline{n_{k-1}}\cdot  (-0.24441406130094) + \overline{n_{k-3}}\cdot  (-4.3045949100653) \\&+ \overline{t_k}\cdot  0.61624905468049 + \overline{t_{k-2}}\cdot  2.1138749574221 + \overline{t_{k-3}}\cdot  -0.32086813690263\\
y&=y-0.018944359148021 \cdot  \cos(s) -0.024568581080264 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.90786223620836) + \overline{v_{k-1}}\cdot  0.25228903489171 + \overline{v_{k-2}}\cdot  (-0.53485124507674) + \overline{v_{k-3}}\cdot  (-1.8522537551352) \\&+ \overline{n_k}\cdot  1.5666089210993 + \overline{n_{k-1}}\cdot  (-0.24489915142453) + \overline{n_{k-3}}\cdot  (-3.8189488996167) \\&+ \overline{t_k}\cdot  0.54909352731943 + \overline{t_{k-2}}\cdot  0.9272179256372 + \overline{t_{k-3}}\cdot  1.8009119972273\\
y&=y+ 0.018516761200688 \cdot  \cos(s) + 0.031299127609342 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.84723127607948 + \overline{v_{k-1}}\cdot  0.050280388584138 + \overline{v_{k-2}}\cdot  0.16693223406574 + \overline{v_{k-3}}\cdot  (-0.60517770461978) \\&+ \overline{n_k}\cdot  (-0.79132263098257) + \overline{n_{k-1}}\cdot  0.084043948902495 + \overline{n_{k-3}}\cdot  0.55685759728236 \\&+ \overline{t_k}\cdot  0.34486591580833 + \overline{t_{k-2}}\cdot  1.4416809309088 + \overline{t_{k-3}}\cdot  -0.53320421692272\\
y&=y-3.4990059374596 \cdot  \cos(s) + 0.68253876032375 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.19681773285744) + \overline{v_{k-1}}\cdot  0.058254908022228 + \overline{v_{k-2}}\cdot  (-0.42707928242563) + \overline{v_{k-3}}\cdot  0.10819564756371 \\&+ \overline{n_k}\cdot  0.04704308876312 + \overline{n_{k-1}}\cdot  0.086127140665846 + \overline{n_{k-3}}\cdot  (-0.024740149959336) \\&+ \overline{t_k}\cdot  0.93632616781613 + \overline{t_{k-2}}\cdot  (-0.035165710684237) + \overline{t_{k-3}}\cdot  0.6569247940928\\
y&=y-0.17115042174198 \cdot  \cos(s) -0.28252973516886 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.0019037079266071) + \overline{v_{k-1}}\cdot  0.18580405652159 + \overline{v_{k-2}}\cdot  0.25080372029813 + \overline{v_{k-3}}\cdot  (-0.38404653550615) \\&+ \overline{n_k}\cdot  (-2.6288831462221) + \overline{n_{k-1}}\cdot  (-0.2747417045966) + \overline{n_{k-3}}\cdot  2.1320451793595 \\&+ \overline{t_k}\cdot  (-1.045444967573) + \overline{t_{k-2}}\cdot  (-0.79282650529856) + \overline{t_{k-3}}\cdot  0.031010778657905\\
y&=y+ 0.059811477926127 \cdot  \cos(s) + 0.15289618520218 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.19814051576479 + \overline{v_{k-1}}\cdot  0.12473129275844 + \overline{v_{k-2}}\cdot  0.18289812163033 + \overline{v_{k-3}}\cdot  (-0.040240761962992) \\&+ \overline{n_k}\cdot  (-3.3405395082573) + \overline{n_{k-1}}\cdot  0.11291153891139 + \overline{n_{k-3}}\cdot  (-2.1744195857812) \\&+ \overline{t_k}\cdot  0.16544111454727 + \overline{t_{k-2}}\cdot  1.0327303718767 + \overline{t_{k-3}}\cdot  1.5923228862532\\
y&=y-0.010118619494568 \cdot  \cos(s) -0.0074420349679962 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.11678811433869) + \overline{v_{k-1}}\cdot  0.40545196498035 + \overline{v_{k-2}}\cdot  (-0.32209724522756) + \overline{v_{k-3}}\cdot  (-1.2191020365812) \\&+ \overline{n_k}\cdot  (-7.0124770086352) + \overline{n_{k-1}}\cdot  0.03675927486414 + \overline{n_{k-3}}\cdot  5.5603653868838 \\&+ \overline{t_k}\cdot  0.37164881797739 + \overline{t_{k-2}}\cdot  (-1.5113345858393) + \overline{t_{k-3}}\cdot  0.66274581255803\\
y&=y+ 0.086117006049066 \cdot  \cos(s) -0.04913820310423 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.32593022105524) + \overline{v_{k-1}}\cdot  0.079875980525919 + \overline{v_{k-2}}\cdot  0.099870348032266 + \overline{v_{k-3}}\cdot  (-0.67291405487035) \\&+ \overline{n_k}\cdot  (-0.37666675860489) + \overline{n_{k-1}}\cdot  0.20938689455095 + \overline{n_{k-3}}\cdot  9.1755464525272 \\&+ \overline{t_k}\cdot  2.2122935682262 + \overline{t_{k-2}}\cdot  0.16197494065924 + \overline{t_{k-3}}\cdot  -0.018643916230819\\
y&=y+ 0.0022606327417702 \cdot  \cos(s) + 0.023805375609675 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.84460551424592 + \overline{v_{k-1}}\cdot  (-0.33094975587693) + \overline{v_{k-2}}\cdot  (-0.010826144353245) + \overline{v_{k-3}}\cdot  0.27948211496192 \\&+ \overline{n_k}\cdot  0.17197033871996 + \overline{n_{k-1}}\cdot  (-0.097541935974101) + \overline{n_{k-3}}\cdot  (-0.30621259089848) \\&+ \overline{t_k}\cdot  (-0.35493284958512) + \overline{t_{k-2}}\cdot  (-0.1698519033907) + \overline{t_{k-3}}\cdot  0.70620227752212\\
y&=y-0.053636933766471 \cdot  \cos(s) + 0.40565361766112 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.17876901517902 + \overline{v_{k-1}}\cdot  0.070199320612956 + \overline{v_{k-2}}\cdot  0.14344749513824 + \overline{v_{k-3}}\cdot  (-0.46963022472564) \\&+ \overline{n_k}\cdot  6.3748083838038 + \overline{n_{k-1}}\cdot  0.11207349357088 + \overline{n_{k-3}}\cdot  (-4.1269050756988) \\&+ \overline{t_k}\cdot  0.63174216577317 + \overline{t_{k-2}}\cdot  (-0.10275316910193) + \overline{t_{k-3}}\cdot  -0.1838974495232\\
y&=y+ 0.0091583716548813 \cdot  \cos(s) + 0.064570155156537 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.13173338939796 + \overline{v_{k-1}}\cdot  (-0.28009532216854) + \overline{v_{k-2}}\cdot  0.092772965987904 + \overline{v_{k-3}}\cdot  (-1.5059798078413) \\&+ \overline{n_k}\cdot  (-2.1352598181993) + \overline{n_{k-1}}\cdot  0.10509416454745 + \overline{n_{k-3}}\cdot  (-0.53606803801543) \\&+ \overline{t_k}\cdot  (-0.18990485591095) + \overline{t_{k-2}}\cdot  0.77387432217994 + \overline{t_{k-3}}\cdot  -0.23630436139784\\
y&=y+ 0.024383694318798 \cdot  \cos(s) -0.073684695482718 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.8804356524727 + \overline{v_{k-1}}\cdot  (-0.14927153177112) + \overline{v_{k-2}}\cdot  (-0.057756555652038) + \overline{v_{k-3}}\cdot  1.0075676397375 \\&+ \overline{n_k}\cdot  (-2.4857722942484) + \overline{n_{k-1}}\cdot  0.15649829748824 + \overline{n_{k-3}}\cdot  2.5030070853474 \\&+ \overline{t_k}\cdot  1.3304923152072 + \overline{t_{k-2}}\cdot  (-0.12114833963005) + \overline{t_{k-3}}\cdot  -0.31746502804656\\
y&=y+ 0.084057404036463 \cdot  \cos(s) + 0.080342220979079 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.45892995128513) + \overline{v_{k-1}}\cdot  (-0.087896961270462) + \overline{v_{k-2}}\cdot  (-0.28713116616515) + \overline{v_{k-3}}\cdot  (-0.37405851179436) \\&+ \overline{n_k}\cdot  (-1.6627204298282) + \overline{n_{k-1}}\cdot  (-0.10621951132626) + \overline{n_{k-3}}\cdot  (-0.19502828709312) \\&+ \overline{t_k}\cdot  (-0.088310579759066) + \overline{t_{k-2}}\cdot  (-1.2096077751772) + \overline{t_{k-3}}\cdot  1.2425887574684\\
y&=y-0.10632114607956 \cdot  \cos(s) + 0.028028588569602 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.31768351305308 + \overline{v_{k-1}}\cdot  0.060754198486302 + \overline{v_{k-2}}\cdot  0.013545521015915 + \overline{v_{k-3}}\cdot  (-0.7766269370097) \\&+ \overline{n_k}\cdot  (-0.61787733817351) + \overline{n_{k-1}}\cdot  0.10467080505564 + \overline{n_{k-3}}\cdot  1.2082028035424 \\&+ \overline{t_k}\cdot  (-0.37850368236208) + \overline{t_{k-2}}\cdot  0.4094129670629 + \overline{t_{k-3}}\cdot  -0.53695066829666\\
y&=y+ 0.43252746651226 \cdot  \cos(s) + 0.016552122636672 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.25758237719377 + \overline{v_{k-1}}\cdot  (-0.25356195272768) + \overline{v_{k-2}}\cdot  (-0.6786377277186) + \overline{v_{k-3}}\cdot  (-1.0718851905902) \\&+ \overline{n_k}\cdot  0.83322277658286 + \overline{n_{k-1}}\cdot  (-0.30758875879175) + \overline{n_{k-3}}\cdot  (-0.63470396094589) \\&+ \overline{t_k}\cdot  1.2643960155815 + \overline{t_{k-2}}\cdot  (-1.0684094036204) + \overline{t_{k-3}}\cdot  0.45116263179746\\
y&=y-0.0074853421942671 \cdot  \cos(s) -0.068405137339272 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.26216784296152) + \overline{v_{k-1}}\cdot  (-0.41730575070274) + \overline{v_{k-2}}\cdot  0.015155508164056 + \overline{v_{k-3}}\cdot  (-0.24359833967796) \\&+ \overline{n_k}\cdot  (-0.12071617316971) + \overline{n_{k-1}}\cdot  (-0.16570655132297) + \overline{n_{k-3}}\cdot  (-1.7909030170594) \\&+ \overline{t_k}\cdot  (-0.40755201770749) + \overline{t_{k-2}}\cdot  (-1.0229730332706) + \overline{t_{k-3}}\cdot  -0.68446950944111\\
y&=y-0.049061333210103 \cdot  \cos(s) -0.18136933353012 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.17677316792512) + \overline{v_{k-1}}\cdot  (-0.025469605233643) + \overline{v_{k-2}}\cdot  (-0.1910114792626) + \overline{v_{k-3}}\cdot  0.92943887248222 \\&+ \overline{n_k}\cdot  0.35620276653333 + \overline{n_{k-1}}\cdot  (-0.23245634977757) + \overline{n_{k-3}}\cdot  (-1.8916035502678) \\&+ \overline{t_k}\cdot  0.74548920174067 + \overline{t_{k-2}}\cdot  (-0.40759478019872) + \overline{t_{k-3}}\cdot  -1.1381431030634\\
y&=y-0.016909584800276 \cdot  \cos(s) + 0.10722899471275 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.43977970353579) + \overline{v_{k-1}}\cdot  (-0.15622561448743) + \overline{v_{k-2}}\cdot  0.64811378783201 + \overline{v_{k-3}}\cdot  (-0.14286146577477) \\&+ \overline{n_k}\cdot  0.66032866159992 + \overline{n_{k-1}}\cdot  (-0.29121017753958) + \overline{n_{k-3}}\cdot  (-2.2201938034659) \\&+ \overline{t_k}\cdot  0.22837161242404 + \overline{t_{k-2}}\cdot  1.0452005681998 + \overline{t_{k-3}}\cdot  2.7703274410098\\
y&=y+ 0.047939907064817 \cdot  \cos(s) + 0.058542345464866 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.80878100421884 + \overline{v_{k-1}}\cdot  (-0.1832152414184) + \overline{v_{k-2}}\cdot  0.79808714115796 + \overline{v_{k-3}}\cdot  0.32102197337873 \\&+ \overline{n_k}\cdot  0.30887569585679 + \overline{n_{k-1}}\cdot  (-0.019982765911454) + \overline{n_{k-3}}\cdot  (-0.048597787228745) \\&+ \overline{t_k}\cdot  0.83984658398251 + \overline{t_{k-2}}\cdot  0.095459155459479 + \overline{t_{k-3}}\cdot  -1.5308996912961\\
y&=y-0.027583794700889 \cdot  \cos(s) -0.015807980686177 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.75135425825118 + \overline{v_{k-1}}\cdot  0.32907884453582 + \overline{v_{k-2}}\cdot  0.083193143066692 + \overline{v_{k-3}}\cdot  0.63183684850071 \\&+ \overline{n_k}\cdot  1.453284313977 + \overline{n_{k-1}}\cdot  (-0.043615950135482) + \overline{n_{k-3}}\cdot  (-0.91140517823509) \\&+ \overline{t_k}\cdot  0.38237529105443 + \overline{t_{k-2}}\cdot  0.32872090921479 + \overline{t_{k-3}}\cdot  -0.12651777143292\\
y&=y-0.051786986817071 \cdot  \cos(s) -0.082560931826909 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.011668877891523 + \overline{v_{k-1}}\cdot  (-0.041078264870397) + \overline{v_{k-2}}\cdot  0.62851420706784 + \overline{v_{k-3}}\cdot  (-0.6894866112561) \\&+ \overline{n_k}\cdot  4.157527054699 + \overline{n_{k-1}}\cdot  (-0.16842535817262) + \overline{n_{k-3}}\cdot  (-4.8078863734826) \\&+ \overline{t_k}\cdot  (-0.34741726477873) + \overline{t_{k-2}}\cdot  (-0.65921610281342) + \overline{t_{k-3}}\cdot  0.86790778494934\\
y&=y-0.084121464323058 \cdot  \cos(s) -0.11677906289896 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.46558466749701) + \overline{v_{k-1}}\cdot  (-0.18491041977008) + \overline{v_{k-2}}\cdot  (-0.17118747295036) + \overline{v_{k-3}}\cdot  0.49679229476403 \\&+ \overline{n_k}\cdot  2.5697896940838 + \overline{n_{k-1}}\cdot  (-0.066126179601844) + \overline{n_{k-3}}\cdot  (-0.82627814981748) \\&+ \overline{t_k}\cdot  0.31777498840923 + \overline{t_{k-2}}\cdot  0.84316776324651 + \overline{t_{k-3}}\cdot  0.95729808447415\\
y&=y-0.11881647098682 \cdot  \cos(s) -0.0028600422974877 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  1.6037567163433 + \overline{v_{k-1}}\cdot  0.00053758943283798 + \overline{v_{k-2}}\cdot  (-0.32091470765556) + \overline{v_{k-3}}\cdot  0.97084640098594 \\&+ \overline{n_k}\cdot  (-2.9941574018056) + \overline{n_{k-1}}\cdot  (-0.46712568401221) + \overline{n_{k-3}}\cdot  (-1.8116137105656) \\&+ \overline{t_k}\cdot  0.054856490528935 + \overline{t_{k-2}}\cdot  0.48579759317903 + \overline{t_{k-3}}\cdot  -0.24774637606093\\
y&=y+ 0.015369878517688 \cdot  \cos(s) -0.0014170875474372 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.23987681510562) + \overline{v_{k-1}}\cdot  (-0.25593431278915) + \overline{v_{k-2}}\cdot  0.0019787299505214 + \overline{v_{k-3}}\cdot  (-0.32541985863337) \\&+ \overline{n_k}\cdot  (-2.9560371696221) + \overline{n_{k-1}}\cdot  (-0.34906715633717) + \overline{n_{k-3}}\cdot  (-3.7363276022986) \\&+ \overline{t_k}\cdot  0.042673616217396 + \overline{t_{k-2}}\cdot  (-0.58697902541369) + \overline{t_{k-3}}\cdot  -0.7759755094146\\
y&=y-0.015786717443258 \cdot  \cos(s) + 0.024975094967218 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.22583019727704) + \overline{v_{k-1}}\cdot  (-0.077395441495797) + \overline{v_{k-2}}\cdot  (-0.90070097121186) + \overline{v_{k-3}}\cdot  (-1.3515781775198) \\&+ \overline{n_k}\cdot  0.96817936231753 + \overline{n_{k-1}}\cdot  0.016511460088508 + \overline{n_{k-3}}\cdot  (-0.80685627363305) \\&+ \overline{t_k}\cdot  (-1.4141167726755) + \overline{t_{k-2}}\cdot  (-0.14414186371363) + \overline{t_{k-3}}\cdot  -1.1413811739829\\
y&=y+ 0.03439862320363 \cdot  \cos(s) -0.046598080481382 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.36159338585778 + \overline{v_{k-1}}\cdot  0.024673785898007 + \overline{v_{k-2}}\cdot  (-0.49530279421355) + \overline{v_{k-3}}\cdot  (-0.53200062724239) \\&+ \overline{n_k}\cdot  4.7428418297513 + \overline{n_{k-1}}\cdot  (-0.13520534322518) + \overline{n_{k-3}}\cdot  2.7194971849481 \\&+ \overline{t_k}\cdot  (-0.34157058625228) + \overline{t_{k-2}}\cdot  (-0.14560613302904) + \overline{t_{k-3}}\cdot  -1.3550265164725\\
y&=y-0.011518035023694 \cdot  \cos(s) -0.053273940541563 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.43925709965552 + \overline{v_{k-1}}\cdot  0.045770084803915 + \overline{v_{k-2}}\cdot  (-0.28623521155044) + \overline{v_{k-3}}\cdot  (-0.36182542785672) \\&+ \overline{n_k}\cdot  1.8435442636642 + \overline{n_{k-1}}\cdot  0.13764671627122 + \overline{n_{k-3}}\cdot  (-1.7369880310385) \\&+ \overline{t_k}\cdot  (-0.67718098088346) + \overline{t_{k-2}}\cdot  0.96181819307071 + \overline{t_{k-3}}\cdot  0.90188716127572\\
y&=y+ 0.23288826713576 \cdot  \cos(s) -0.15672951801364 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.064598397316458 + \overline{v_{k-1}}\cdot  (-0.096604519115184) + \overline{v_{k-2}}\cdot  (-0.39600241189883) + \overline{v_{k-3}}\cdot  0.82081540685491 \\&+ \overline{n_k}\cdot  3.7967918536811 + \overline{n_{k-1}}\cdot  (-0.32479462323413) + \overline{n_{k-3}}\cdot  (-6.8163120661345) \\&+ \overline{t_k}\cdot  (-0.66256830456833) + \overline{t_{k-2}}\cdot  (-0.72268349097057) + \overline{t_{k-3}}\cdot  -0.60484037951159\\
y&=y-0.0046388898142893 \cdot  \cos(s) -0.079687036388495 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.37445329094819 + \overline{v_{k-1}}\cdot  (-0.363050810254) + \overline{v_{k-2}}\cdot  (-0.85538049193332) + \overline{v_{k-3}}\cdot  0.52346497545624 \\&+ \overline{n_k}\cdot  4.1858898231664 + \overline{n_{k-1}}\cdot  (-0.15817295606714) + \overline{n_{k-3}}\cdot  (-6.0893988831619) \\&+ \overline{t_k}\cdot  0.45630187667385 + \overline{t_{k-2}}\cdot  (-0.074689397657911) + \overline{t_{k-3}}\cdot  -0.44773066097934\\
y&=y+ 0.044154822592607 \cdot  \cos(s) -0.13794930346985 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.059317512967557) + \overline{v_{k-1}}\cdot  (-0.11022537799313) + \overline{v_{k-2}}\cdot  0.31400317452795 + \overline{v_{k-3}}\cdot  (-0.75274374233922) \\&+ \overline{n_k}\cdot  1.0373155190017 + \overline{n_{k-1}}\cdot  0.30714349808628 + \overline{n_{k-3}}\cdot  (-3.5554839662073) \\&+ \overline{t_k}\cdot  (-1.2221273756119) + \overline{t_{k-2}}\cdot  0.50864317961329 + \overline{t_{k-3}}\cdot  -0.84515229970508\\
y&=y-0.060686607415732 \cdot  \cos(s) -0.088015554200766 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.29244085118027) + \overline{v_{k-1}}\cdot  (-0.019183333270204) + \overline{v_{k-2}}\cdot  (-0.077483622043874) + \overline{v_{k-3}}\cdot  (-1.0736630167229) \\&+ \overline{n_k}\cdot  3.04419549621 + \overline{n_{k-1}}\cdot  (-0.11645011564443) + \overline{n_{k-3}}\cdot  (-3.7448151043279) \\&+ \overline{t_k}\cdot  (-1.0090715552592) + \overline{t_{k-2}}\cdot  (-2.0064460721785) + \overline{t_{k-3}}\cdot  -1.0847868318571\\
y&=y+ 0.064300654108802 \cdot  \cos(s) + 0.025756630419533 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.20500545105685 + \overline{v_{k-1}}\cdot  0.042721643850222 + \overline{v_{k-2}}\cdot  (-0.28239556672529) + \overline{v_{k-3}}\cdot  0.029533614880545 \\&+ \overline{n_k}\cdot  (-0.50155646564908) + \overline{n_{k-1}}\cdot  (-0.17524499316815) + \overline{n_{k-3}}\cdot  5.6667285838448 \\&+ \overline{t_k}\cdot  0.24892316862687 + \overline{t_{k-2}}\cdot  (-0.57378696284797) + \overline{t_{k-3}}\cdot  1.6207438609699\\
y&=y+ 0.053651372132957 \cdot  \cos(s) -0.019255348978032 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.80835434001989 + \overline{v_{k-1}}\cdot  0.033730865058327 + \overline{v_{k-2}}\cdot  (-0.36192000465059) + \overline{v_{k-3}}\cdot  0.068972098703966 \\&+ \overline{n_k}\cdot  0.39392332382431 + \overline{n_{k-1}}\cdot  0.02545045216391 + \overline{n_{k-3}}\cdot  1.8776375750048 \\&+ \overline{t_k}\cdot  0.84579939266458 + \overline{t_{k-2}}\cdot  (-0.83045785788514) + \overline{t_{k-3}}\cdot  -0.48607999231105\\
y&=y-0.11951754174837 \cdot  \cos(s) -0.015213137309564 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.61621830446052 + \overline{v_{k-1}}\cdot  (-0.20213981234134) + \overline{v_{k-2}}\cdot  (-0.68587765032013) + \overline{v_{k-3}}\cdot  (-0.16699261687744) \\&+ \overline{n_k}\cdot  0.57880370456389 + \overline{n_{k-1}}\cdot  (-0.097281575761222) + \overline{n_{k-3}}\cdot  (-0.10123488308088) \\&+ \overline{t_k}\cdot  (-0.65262972664406) + \overline{t_{k-2}}\cdot  (-0.77807867134149) + \overline{t_{k-3}}\cdot  0.43874433329463\\
y&=y+ 0.67897256508403 \cdot  \cos(s) + 0.45842360791261 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.69715873877587 + \overline{v_{k-1}}\cdot  (-0.066252785620862) + \overline{v_{k-2}}\cdot  0.27855742844366 + \overline{v_{k-3}}\cdot  (-0.72016847115053) \\&+ \overline{n_k}\cdot  9.004546581274 + \overline{n_{k-1}}\cdot  (-0.023021945093869) + \overline{n_{k-3}}\cdot  (-8.0358050609837) \\&+ \overline{t_k}\cdot  0.070269407451441 + \overline{t_{k-2}}\cdot  (-0.17145826222352) + \overline{t_{k-3}}\cdot  0.23371762786191\\
y&=y-0.11558159654724 \cdot  \cos(s) + 0.1280981152627 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.061123532802879 + \overline{v_{k-1}}\cdot  0.31117935807833 + \overline{v_{k-2}}\cdot  (-0.042175847334849) + \overline{v_{k-3}}\cdot  1.0643277928174 \\&+ \overline{n_k}\cdot  3.571742118105 + \overline{n_{k-1}}\cdot  0.060501661631117 + \overline{n_{k-3}}\cdot  (-3.1048698462113) \\&+ \overline{t_k}\cdot  0.40682378280127 + \overline{t_{k-2}}\cdot  0.83377672510518 + \overline{t_{k-3}}\cdot  -1.0219143195115\\
y&=y+ 0.12748200297507 \cdot  \cos(s) + 0.049135256473724 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.44209301626502) + \overline{v_{k-1}}\cdot  0.054516649430299 + \overline{v_{k-2}}\cdot  (-0.71758192434574) + \overline{v_{k-3}}\cdot  0.69285987658271 \\&+ \overline{n_k}\cdot  (-1.9573420448319) + \overline{n_{k-1}}\cdot  0.37990622043178 + \overline{n_{k-3}}\cdot  (-2.4086154903417) \\&+ \overline{t_k}\cdot  (-0.31509319788408) + \overline{t_{k-2}}\cdot  0.91215670154238 + \overline{t_{k-3}}\cdot  -0.40144522153246\\
y&=y-0.034712965813719 \cdot  \cos(s) + 0.046483858820691 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.8115922537416) + \overline{v_{k-1}}\cdot  0.015683113181554 + \overline{v_{k-2}}\cdot  0.041448036256392 + \overline{v_{k-3}}\cdot  (-0.58974637492163) \\&+ \overline{n_k}\cdot  1.0023351224226 + \overline{n_{k-1}}\cdot  (-0.10293406857145) + \overline{n_{k-3}}\cdot  (-0.81923159078529) \\&+ \overline{t_k}\cdot  (-1.0525319967428) + \overline{t_{k-2}}\cdot  (-0.4659667165728) + \overline{t_{k-3}}\cdot  -1.2676270486531\\
y&=y-0.068291348633799 \cdot  \cos(s) + 0.077118109718707 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.6870714685788) + \overline{v_{k-1}}\cdot  (-0.16954182644662) + \overline{v_{k-2}}\cdot  (-0.15518803300695) + \overline{v_{k-3}}\cdot  (-0.10882766625271) \\&+ \overline{n_k}\cdot  0.95457462902051 + \overline{n_{k-1}}\cdot  (-0.23744205001861) + \overline{n_{k-3}}\cdot  (-0.36616055202298) \\&+ \overline{t_k}\cdot  0.3541862497669 + \overline{t_{k-2}}\cdot  0.057071813041939 + \overline{t_{k-3}}\cdot  -0.18828258229724\\
y&=y-0.12766999274979 \cdot  \cos(s) + 0.37594579379313 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-1.2233133350886) + \overline{v_{k-1}}\cdot  0.2818169111056 + \overline{v_{k-2}}\cdot  0.68768773694539 + \overline{v_{k-3}}\cdot  0.30228449129992 \\&+ \overline{n_k}\cdot  2.9491625036004 + \overline{n_{k-1}}\cdot  0.021740132213148 + \overline{n_{k-3}}\cdot  (-0.78981562725561) \\&+ \overline{t_k}\cdot  (-1.1797429142248) + \overline{t_{k-2}}\cdot  (-1.3043266779761) + \overline{t_{k-3}}\cdot  0.27035547879731\\
y&=y-0.0043180756877553 \cdot  \cos(s) -0.039611362195905 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.40396480218111 + \overline{v_{k-1}}\cdot  0.036748442602898 + \overline{v_{k-2}}\cdot  (-0.61937693804835) + \overline{v_{k-3}}\cdot  0.27402193978315 \\&+ \overline{n_k}\cdot  (-1.7842147913316) + \overline{n_{k-1}}\cdot  0.18969225643528 + \overline{n_{k-3}}\cdot  2.5576465009336 \\&+ \overline{t_k}\cdot  (-0.15063584259683) + \overline{t_{k-2}}\cdot  (-2.0517360876039) + \overline{t_{k-3}}\cdot  1.775081107002\\
y&=y+ 0.020784074404751 \cdot  \cos(s) -0.10852650573083 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.4028289598915 + \overline{v_{k-1}}\cdot  (-0.073175411728252) + \overline{v_{k-2}}\cdot  (-0.53458882076387) + \overline{v_{k-3}}\cdot  0.29728563312547 \\&+ \overline{n_k}\cdot  0.62698706643598 + \overline{n_{k-1}}\cdot  (-0.1625781539619) + \overline{n_{k-3}}\cdot  (-0.17510801055693) \\&+ \overline{t_k}\cdot  0.51199125506705 + \overline{t_{k-2}}\cdot  0.54716019643085 + \overline{t_{k-3}}\cdot  -1.3632242063645\\
y&=y+ 0.3427433447271 \cdot  \cos(s) + 0.29556700908644 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.24719249279029 + \overline{v_{k-1}}\cdot  0.53888175734965 + \overline{v_{k-2}}\cdot  0.82640950274578 + \overline{v_{k-3}}\cdot  0.77593890535894 \\&+ \overline{n_k}\cdot  (-1.4871335430835) + \overline{n_{k-1}}\cdot  0.063923384775276 + \overline{n_{k-3}}\cdot  (-1.4823186298831) \\&+ \overline{t_k}\cdot  0.45828293578681 + \overline{t_{k-2}}\cdot  0.61353634209822 + \overline{t_{k-3}}\cdot  -0.030632200465192\\
y&=y-0.079967881764442 \cdot  \cos(s) -0.039971495234953 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.012422692796249 + \overline{v_{k-1}}\cdot  0.28588048766519 + \overline{v_{k-2}}\cdot  (-0.14705908711552) + \overline{v_{k-3}}\cdot  1.0347973283625 \\&+ \overline{n_k}\cdot  (-3.0289662397224) + \overline{n_{k-1}}\cdot  (-0.21657919993359) + \overline{n_{k-3}}\cdot  (-3.0576659047943) \\&+ \overline{t_k}\cdot  (-0.52152583763585) + \overline{t_{k-2}}\cdot  0.60779451482956 + \overline{t_{k-3}}\cdot  1.5724457442509\\
y&=y+ 0.016197643889979 \cdot  \cos(s) -0.02311509558311 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.61141511700201) + \overline{v_{k-1}}\cdot  0.13290626223625 + \overline{v_{k-2}}\cdot  0.095219977256908 + \overline{v_{k-3}}\cdot  (-0.14657309929045) \\&+ \overline{n_k}\cdot  (-4.1277646553299) + \overline{n_{k-1}}\cdot  0.11068453286229 + \overline{n_{k-3}}\cdot  2.8747029628855 \\&+ \overline{t_k}\cdot  (-0.54215387567982) + \overline{t_{k-2}}\cdot  (-0.45116706073738) + \overline{t_{k-3}}\cdot  1.7741546185254\\
y&=y-0.040138849880116 \cdot  \cos(s) + 0.074962524125227 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.39107551291792) + \overline{v_{k-1}}\cdot  0.12274660608723 + \overline{v_{k-2}}\cdot  (-0.60178277343316) + \overline{v_{k-3}}\cdot  0.6602659147272 \\&+ \overline{n_k}\cdot  1.5156204231173 + \overline{n_{k-1}}\cdot  (-0.010184729966952) + \overline{n_{k-3}}\cdot  (-1.8673434534205) \\&+ \overline{t_k}\cdot  0.28893976397337 + \overline{t_{k-2}}\cdot  (-0.77075294468846) + \overline{t_{k-3}}\cdot  -0.19924946508226\\
y&=y-0.20336766861947 \cdot  \cos(s) + 0.33595325035672 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.013413509884823 + \overline{v_{k-1}}\cdot  0.040304867605146 + \overline{v_{k-2}}\cdot  0.79704844922216 + \overline{v_{k-3}}\cdot  0.83154542377705 \\&+ \overline{n_k}\cdot  0.14268370176646 + \overline{n_{k-1}}\cdot  0.11559116505276 + \overline{n_{k-3}}\cdot  1.9640395532104 \\&+ \overline{t_k}\cdot  1.38891577714 + \overline{t_{k-2}}\cdot  0.34927172570989 + \overline{t_{k-3}}\cdot  0.46568732037223\\
y&=y+ 0.069177423378979 \cdot  \cos(s) -0.061434451343931 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.23627975818342 + \overline{v_{k-1}}\cdot  0.026179118145544 + \overline{v_{k-2}}\cdot  (-0.43496460935533) + \overline{v_{k-3}}\cdot  (-0.2957285364787) \\&+ \overline{n_k}\cdot  0.8359053206001 + \overline{n_{k-1}}\cdot  (-0.13529122795089) + \overline{n_{k-3}}\cdot  (-0.59076809851459) \\&+ \overline{t_k}\cdot  (-0.31742811991633) + \overline{t_{k-2}}\cdot  (-1.5477420509238) + \overline{t_{k-3}}\cdot  0.5490837993564\\
y&=y+ 3.0717135720378 \cdot  \cos(s) + 0.42595342511833 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.33577157755192) + \overline{v_{k-1}}\cdot  (-0.1890541145897) + \overline{v_{k-2}}\cdot  (-0.044115794840124) + \overline{v_{k-3}}\cdot  0.25836953839076 \\&+ \overline{n_k}\cdot  2.6503599150371 + \overline{n_{k-1}}\cdot  (-0.18131623386889) + \overline{n_{k-3}}\cdot  3.0719027772353 \\&+ \overline{t_k}\cdot  0.24558608715437 + \overline{t_{k-2}}\cdot  0.90474113435491 + \overline{t_{k-3}}\cdot  -1.4636261033049\\
y&=y+ 0.025265155820038 \cdot  \cos(s) + 0.0082123072262535 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.25459319593585 + \overline{v_{k-1}}\cdot  (-0.388183457583) + \overline{v_{k-2}}\cdot  (-0.57560753449426) + \overline{v_{k-3}}\cdot  0.076838701081425 \\&+ \overline{n_k}\cdot  2.100720398664 + \overline{n_{k-1}}\cdot  (-0.099082438310629) + \overline{n_{k-3}}\cdot  0.58133229208003 \\&+ \overline{t_k}\cdot  (-0.016242740362897) + \overline{t_{k-2}}\cdot  1.1969145539546 + \overline{t_{k-3}}\cdot  -2.6154766617621\\
y&=y-0.055206719748503 \cdot  \cos(s) -0.037797398860482 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.25188182923633) + \overline{v_{k-1}}\cdot  (-0.18007007905242) + \overline{v_{k-2}}\cdot  (-0.36475152983176) + \overline{v_{k-3}}\cdot  (-0.087203328561246) \\&+ \overline{n_k}\cdot  0.48076813999135 + \overline{n_{k-1}}\cdot  (-0.17766311279376) + \overline{n_{k-3}}\cdot  1.688345899356 \\&+ \overline{t_k}\cdot  (-0.096706131852119) + \overline{t_{k-2}}\cdot  (-0.14158435712413) + \overline{t_{k-3}}\cdot  -0.29487491948705\\
y&=y-0.03430882993683 \cdot  \cos(s) + 0.15894107821953 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.40576220232117 + \overline{v_{k-1}}\cdot  0.080362129537167 + \overline{v_{k-2}}\cdot  0.27772777647952 + \overline{v_{k-3}}\cdot  0.4482599293108 \\&+ \overline{n_k}\cdot  2.2325772254501 + \overline{n_{k-1}}\cdot  (-0.13769785324422) + \overline{n_{k-3}}\cdot  (-3.8766768620419) \\&+ \overline{t_k}\cdot  (-0.041222843424161) + \overline{t_{k-2}}\cdot  (-0.27890131495352) + \overline{t_{k-3}}\cdot  0.19552807997064\\
y&=y+ 0.040853423247748 \cdot  \cos(s) + 0.026081877370598 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  1.0486970586256 + \overline{v_{k-1}}\cdot  0.58480202365134 + \overline{v_{k-2}}\cdot  0.19761511597858 + \overline{v_{k-3}}\cdot  (-0.48318636514403) \\&+ \overline{n_k}\cdot  2.1692622059488 + \overline{n_{k-1}}\cdot  0.104324832837 + \overline{n_{k-3}}\cdot  3.0013041490242 \\&+ \overline{t_k}\cdot  (-0.0013034081387521) + \overline{t_{k-2}}\cdot  (-0.2755002819141) + \overline{t_{k-3}}\cdot  -1.5566669526027\\
y&=y-0.0013882997029764 \cdot  \cos(s) -0.015511807781405 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.31386081002059) + \overline{v_{k-1}}\cdot  0.1293020499412 + \overline{v_{k-2}}\cdot  0.11529795460325 + \overline{v_{k-3}}\cdot  (-0.24822392526695) \\&+ \overline{n_k}\cdot  (-3.3160561798488) + \overline{n_{k-1}}\cdot  0.44706491421305 + \overline{n_{k-3}}\cdot  (-1.8294190352312) \\&+ \overline{t_k}\cdot  (-0.053453033705414) + \overline{t_{k-2}}\cdot  (-0.49280245287429) + \overline{t_{k-3}}\cdot  0.19694055980245\\
y&=y-0.060980637003501 \cdot  \cos(s) + 0.14055568329736 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.25994500139995 + \overline{v_{k-1}}\cdot  (-0.078939580699695) + \overline{v_{k-2}}\cdot  (-0.01494390554366) + \overline{v_{k-3}}\cdot  0.47595755604079 \\&+ \overline{n_k}\cdot  0.81351882927992 + \overline{n_{k-1}}\cdot  0.071500271326471 + \overline{n_{k-3}}\cdot  (-0.55798198062438) \\&+ \overline{t_k}\cdot  (-1.1381046703675) + \overline{t_{k-2}}\cdot  (-1.0426145502673) + \overline{t_{k-3}}\cdot  -0.59339919002787\\
y&=y-0.089029068082855 \cdot  \cos(s) + 0.004576124468916 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.73157058341543) + \overline{v_{k-1}}\cdot  0.3700308214291 + \overline{v_{k-2}}\cdot  0.33500770801749 + \overline{v_{k-3}}\cdot  0.018024187931452 \\&+ \overline{n_k}\cdot  1.2205331546415 + \overline{n_{k-1}}\cdot  (-0.18169028160815) + \overline{n_{k-3}}\cdot  (-4.4857174787933) \\&+ \overline{t_k}\cdot  (-0.75684286937228) + \overline{t_{k-2}}\cdot  0.34485953827102 + \overline{t_{k-3}}\cdot  0.085304783163633\\
y&=y+ 0.085377058203667 \cdot  \cos(s) + 0.031528621633092 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.15242786643233) + \overline{v_{k-1}}\cdot  0.0025867522449353 + \overline{v_{k-2}}\cdot  0.44500713480761 + \overline{v_{k-3}}\cdot  (-0.79318481758384) \\&+ \overline{n_k}\cdot  (-1.7823408419956) + \overline{n_{k-1}}\cdot  0.20269196179305 + \overline{n_{k-3}}\cdot  (-4.8894343557039) \\&+ \overline{t_k}\cdot  0.65510375537264 + \overline{t_{k-2}}\cdot  0.80417195998693 + \overline{t_{k-3}}\cdot  0.8577866112518\\
y&=y+ 0.014358954022024 \cdot  \cos(s) + 0.029772391811455 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.063928130376962) + \overline{v_{k-1}}\cdot  (-0.0046039198715692) + \overline{v_{k-2}}\cdot  0.67747500772939 + \overline{v_{k-3}}\cdot  0.43190885676114 \\&+ \overline{n_k}\cdot  0.91902505388909 + \overline{n_{k-1}}\cdot  0.20261839326194 + \overline{n_{k-3}}\cdot  (-2.7174511895402) \\&+ \overline{t_k}\cdot  0.62376436068261 + \overline{t_{k-2}}\cdot  (-1.5480884497021) + \overline{t_{k-3}}\cdot  -1.9334713339308\\
y&=y+ 0.00092096943883582 \cdot  \cos(s) + 0.051218065835908 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.17447009697427 + \overline{v_{k-1}}\cdot  0.16264768985038 + \overline{v_{k-2}}\cdot  0.6126307557983 + \overline{v_{k-3}}\cdot  0.38821088563286 \\&+ \overline{n_k}\cdot  1.6460486822933 + \overline{n_{k-1}}\cdot  0.11904629785382 + \overline{n_{k-3}}\cdot  4.4526474812372 \\&+ \overline{t_k}\cdot  (-1.2874473677698) + \overline{t_{k-2}}\cdot  (-0.53353562411878) + \overline{t_{k-3}}\cdot  -0.94954520859861\\
y&=y+ 0.0041032472616334 \cdot  \cos(s) + 0.026064196886314 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.11381273673559) + \overline{v_{k-1}}\cdot  (-0.11309789711385) + \overline{v_{k-2}}\cdot  0.40577937398995 + \overline{v_{k-3}}\cdot  (-0.60012174166095) \\&+ \overline{n_k}\cdot  (-3.0754731732655) + \overline{n_{k-1}}\cdot  0.00059941618785972 + \overline{n_{k-3}}\cdot  (-2.255890362792) \\&+ \overline{t_k}\cdot  (-0.22475924599207) + \overline{t_{k-2}}\cdot  (-0.37080176484437) + \overline{t_{k-3}}\cdot  0.24450911270488\\
y&=y-0.037312990607426 \cdot  \cos(s) -0.13923789452907 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.73311134921818 + \overline{v_{k-1}}\cdot  0.30884254753799 + \overline{v_{k-2}}\cdot  0.19912824792272 + \overline{v_{k-3}}\cdot  0.47867765355674 \\&+ \overline{n_k}\cdot  (-9.9376866939528) + \overline{n_{k-1}}\cdot  (-0.26458859522512) + \overline{n_{k-3}}\cdot  (-3.4585407939256) \\&+ \overline{t_k}\cdot  (-0.64018780183579) + \overline{t_{k-2}}\cdot  0.87941998857369 + \overline{t_{k-3}}\cdot  -1.1364026640412\\
y&=y-0.0080843906837836 \cdot  \cos(s) + 0.0085226826746696 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.77106503579505 + \overline{v_{k-1}}\cdot  (-0.090102588331579) + \overline{v_{k-2}}\cdot  0.18881987091193 + \overline{v_{k-3}}\cdot  0.74168758914324 \\&+ \overline{n_k}\cdot  (-4.6126095589984) + \overline{n_{k-1}}\cdot  0.12580729209152 + \overline{n_{k-3}}\cdot  0.61363369687333 \\&+ \overline{t_k}\cdot  (-0.24884677386098) + \overline{t_{k-2}}\cdot  (-0.65843414165576) + \overline{t_{k-3}}\cdot  -2.2102908294502\\
y&=y+ 0.015989293279952 \cdot  \cos(s) -0.012531332785732 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.32634568173872) + \overline{v_{k-1}}\cdot  0.041571313604241 + \overline{v_{k-2}}\cdot  (-0.76226551693684) + \overline{v_{k-3}}\cdot  (-0.059375361674349) \\&+ \overline{n_k}\cdot  3.5742752522233 + \overline{n_{k-1}}\cdot  (-0.14970045182107) + \overline{n_{k-3}}\cdot  0.97835294350369 \\&+ \overline{t_k}\cdot  0.10905628141524 + \overline{t_{k-2}}\cdot  0.23364474393908 + \overline{t_{k-3}}\cdot  0.94102444425663\\
y&=y-0.0064939405986199 \cdot  \cos(s) -0.046195210898848 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.89739581078345) + \overline{v_{k-1}}\cdot  0.14111364565511 + \overline{v_{k-2}}\cdot  (-0.29049561000264) + \overline{v_{k-3}}\cdot  (-0.23732677579395) \\&+ \overline{n_k}\cdot  0.73573877375747 + \overline{n_{k-1}}\cdot  0.20717826702202 + \overline{n_{k-3}}\cdot  1.0510327826059 \\&+ \overline{t_k}\cdot  (-1.89027936533) + \overline{t_{k-2}}\cdot  0.98497827877588 + \overline{t_{k-3}}\cdot  -1.361749264021\\
y&=y+ 0.040109282095949 \cdot  \cos(s) + 0.034693150867751 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.5826919870097) + \overline{v_{k-1}}\cdot  0.021842676690041 + \overline{v_{k-2}}\cdot  (-0.48892035358605) + \overline{v_{k-3}}\cdot  0.63536266921571 \\&+ \overline{n_k}\cdot  4.1600250518594 + \overline{n_{k-1}}\cdot  0.25507730526919 + \overline{n_{k-3}}\cdot  (-2.2829257029429) \\&+ \overline{t_k}\cdot  (-1.1224303280368) + \overline{t_{k-2}}\cdot  (-0.39902675894711) + \overline{t_{k-3}}\cdot  -1.0826638376451\\
y&=y-0.039981510039405 \cdot  \cos(s) -0.017204146455158 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.030230176956927 + \overline{v_{k-1}}\cdot  (-0.296730661779) + \overline{v_{k-2}}\cdot  0.011517022440393 + \overline{v_{k-3}}\cdot  0.055187508805417 \\&+ \overline{n_k}\cdot  (-1.9566404265983) + \overline{n_{k-1}}\cdot  0.077188842592816 + \overline{n_{k-3}}\cdot  2.068856413021 \\&+ \overline{t_k}\cdot  1.1795985503023 + \overline{t_{k-2}}\cdot  0.68331050991756 + \overline{t_{k-3}}\cdot  2.7860010644792\\
y&=y-0.058584871580974 \cdot  \cos(s) -0.017393943656854 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.31822709333089 + \overline{v_{k-1}}\cdot  (-0.014271464298613) + \overline{v_{k-2}}\cdot  0.70535523626635 + \overline{v_{k-3}}\cdot  0.2742032039563 \\&+ \overline{n_k}\cdot  1.4570613279171 + \overline{n_{k-1}}\cdot  0.11385902494481 + \overline{n_{k-3}}\cdot  (-0.99207722105192) \\&+ \overline{t_k}\cdot  (-0.65912129020136) + \overline{t_{k-2}}\cdot  1.3836058945175 + \overline{t_{k-3}}\cdot  1.7607664501676\\
y&=y+ 0.091013521410747 \cdot  \cos(s) -0.016028891176556 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.86802514838381) + \overline{v_{k-1}}\cdot  (-0.3354593208007) + \overline{v_{k-2}}\cdot  (-0.41552143253572) + \overline{v_{k-3}}\cdot  0.20824900545061 \\&+ \overline{n_k}\cdot  (-3.8462053401617) + \overline{n_{k-1}}\cdot  0.026213222748612 + \overline{n_{k-3}}\cdot  (-0.17350927185547) \\&+ \overline{t_k}\cdot  1.409010967085 + \overline{t_{k-2}}\cdot  0.89354106903164 + \overline{t_{k-3}}\cdot  1.2971490987342\\
y&=y-0.019081474586905 \cdot  \cos(s) -0.022425834954556 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.25793449504941) + \overline{v_{k-1}}\cdot  0.71037864534253 + \overline{v_{k-2}}\cdot  0.7267138173557 + \overline{v_{k-3}}\cdot  1.1998813791212 \\&+ \overline{n_k}\cdot  3.5053884111367 + \overline{n_{k-1}}\cdot  0.14584112687278 + \overline{n_{k-3}}\cdot  (-2.9387848525964) \\&+ \overline{t_k}\cdot  (-0.23244674942071) + \overline{t_{k-2}}\cdot  (-1.4816666350978) + \overline{t_{k-3}}\cdot  -0.16874633991965\\
y&=y+ 0.0095682147175978 \cdot  \cos(s) + 0.011403276459972 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.74100143992648 + \overline{v_{k-1}}\cdot  0.19894178799665 + \overline{v_{k-2}}\cdot  0.018332770745976 + \overline{v_{k-3}}\cdot  (-0.63276674310163) \\&+ \overline{n_k}\cdot  2.3105586916758 + \overline{n_{k-1}}\cdot  (-0.26716792155061) + \overline{n_{k-3}}\cdot  (-2.0023305883282) \\&+ \overline{t_k}\cdot  (-0.044046540936418) + \overline{t_{k-2}}\cdot  (-0.61801259397661) + \overline{t_{k-3}}\cdot  0.019247008483582\\
y&=y+ 0.38594893233202 \cdot  \cos(s) + 0.13289299422207 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.21210027099482) + \overline{v_{k-1}}\cdot  0.23213437753485 + \overline{v_{k-2}}\cdot  0.32295060672409 + \overline{v_{k-3}}\cdot  0.37943237695541 \\&+ \overline{n_k}\cdot  (-0.098498100939365) + \overline{n_{k-1}}\cdot  (-0.06472187290408) + \overline{n_{k-3}}\cdot  (-3.7431422682593) \\&+ \overline{t_k}\cdot  (-1.4681947230847) + \overline{t_{k-2}}\cdot  1.5739407894498 + \overline{t_{k-3}}\cdot  -1.3693647285006\\
y&=y-0.0057533003864996 \cdot  \cos(s) -0.053911558965515 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.3294642883148) + \overline{v_{k-1}}\cdot  (-0.08221291323503) + \overline{v_{k-2}}\cdot  0.21569623682358 + \overline{v_{k-3}}\cdot  (-0.74270301815544) \\&+ \overline{n_k}\cdot  0.19744550085899 + \overline{n_{k-1}}\cdot  (-0.26914920486125) + \overline{n_{k-3}}\cdot  1.3797277456814 \\&+ \overline{t_k}\cdot  (-0.35061993574076) + \overline{t_{k-2}}\cdot  0.65226836204528 + \overline{t_{k-3}}\cdot  0.9871671568106\\
y&=y+ 0.063113280244711 \cdot  \cos(s) + 0.094482739846245 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.0024102825865738) + \overline{v_{k-1}}\cdot  (-0.31836713541293) + \overline{v_{k-2}}\cdot  (-0.034270346647274) + \overline{v_{k-3}}\cdot  0.52430545902216 \\&+ \overline{n_k}\cdot  (-1.6725636427132) + \overline{n_{k-1}}\cdot  0.05855420600357 + \overline{n_{k-3}}\cdot  (-0.33719717795816) \\&+ \overline{t_k}\cdot  0.53398251160395 + \overline{t_{k-2}}\cdot  0.96179910045751 + \overline{t_{k-3}}\cdot  0.762603622521\\
y&=y+ 0.15792444770157 \cdot  \cos(s) -0.022371143214056 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.75413364712181 + \overline{v_{k-1}}\cdot  (-0.45647784695528) + \overline{v_{k-2}}\cdot  1.0265422834213 + \overline{v_{k-3}}\cdot  (-0.68153086045924) \\&+ \overline{n_k}\cdot  (-1.2594124711685) + \overline{n_{k-1}}\cdot  (-0.13599007237109) + \overline{n_{k-3}}\cdot  (-1.7556455791042) \\&+ \overline{t_k}\cdot  (-0.19956850595755) + \overline{t_{k-2}}\cdot  (-0.95035708496566) + \overline{t_{k-3}}\cdot  -0.73805599735145\\
y&=y-0.081925089440202 \cdot  \cos(s) + 0.017141090759093 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.45156317334975) + \overline{v_{k-1}}\cdot  (-0.32357922653324) + \overline{v_{k-2}}\cdot  0.7760892474274 + \overline{v_{k-3}}\cdot  0.43832634996747 \\&+ \overline{n_k}\cdot  3.9279769394259 + \overline{n_{k-1}}\cdot  (-0.11136671902838) + \overline{n_{k-3}}\cdot  (-0.7575971715878) \\&+ \overline{t_k}\cdot  0.37561305045973 + \overline{t_{k-2}}\cdot  0.17832902806532 + \overline{t_{k-3}}\cdot  0.24555793334044\\
y&=y+ 0.13738424777592 \cdot  \cos(s) + 0.053265857701089 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.6906030480701) + \overline{v_{k-1}}\cdot  0.41163068958208 + \overline{v_{k-2}}\cdot  (-0.4027333293875) + \overline{v_{k-3}}\cdot  0.31605385389028 \\&+ \overline{n_k}\cdot  0.076494241743703 + \overline{n_{k-1}}\cdot  (-0.19652125245129) + \overline{n_{k-3}}\cdot  6.2120166079051 \\&+ \overline{t_k}\cdot  (-0.14757603720768) + \overline{t_{k-2}}\cdot  (-1.756674274605) + \overline{t_{k-3}}\cdot  -1.9280803867154\\
y&=y-0.03549032657104 \cdot  \cos(s) + 0.017672908686963 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.48235174914102) + \overline{v_{k-1}}\cdot  (-0.14505745567104) + \overline{v_{k-2}}\cdot  (-0.89285023189899) + \overline{v_{k-3}}\cdot  (-0.65796650618568) \\&+ \overline{n_k}\cdot  (-1.7818085457027) + \overline{n_{k-1}}\cdot  (-0.077077512049044) + \overline{n_{k-3}}\cdot  (-3.1546989700182) \\&+ \overline{t_k}\cdot  0.28593200288056 + \overline{t_{k-2}}\cdot  (-1.7965952265115) + \overline{t_{k-3}}\cdot  0.68987570372033\\
y&=y-0.033383596761382 \cdot  \cos(s) + 0.01708951694821 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.43372466387743) + \overline{v_{k-1}}\cdot  0.0070073398159157 + \overline{v_{k-2}}\cdot  0.28432167179528 + \overline{v_{k-3}}\cdot  0.34117314551252 \\&+ \overline{n_k}\cdot  (-0.91625966586057) + \overline{n_{k-1}}\cdot  0.19353827335367 + \overline{n_{k-3}}\cdot  (-0.24508433580305) \\&+ \overline{t_k}\cdot  (-0.70501247944371) + \overline{t_{k-2}}\cdot  (-1.1859299029664) + \overline{t_{k-3}}\cdot  0.86604144279862\\
y&=y-0.11207783813209 \cdot  \cos(s) -0.066353923098077 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  1.0530682027985 + \overline{v_{k-1}}\cdot  0.28636638293607 + \overline{v_{k-2}}\cdot  0.57150450833751 + \overline{v_{k-3}}\cdot  (-0.11144645862663) \\&+ \overline{n_k}\cdot  (-3.0725638863648) + \overline{n_{k-1}}\cdot  (-0.17342818569404) + \overline{n_{k-3}}\cdot  1.0228359792419 \\&+ \overline{t_k}\cdot  (-0.92256310608875) + \overline{t_{k-2}}\cdot  2.2013540385711 + \overline{t_{k-3}}\cdot  1.2926329931788\\
y&=y-0.030808166862244 \cdot  \cos(s) + 0.038739528019412 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.21742569239359 + \overline{v_{k-1}}\cdot  (-0.30008192867895) + \overline{v_{k-2}}\cdot  0.0090961284191189 + \overline{v_{k-3}}\cdot  0.59998484535416 \\&+ \overline{n_k}\cdot  (-6.836350335213) + \overline{n_{k-1}}\cdot  (-0.24115220856752) + \overline{n_{k-3}}\cdot  (-4.492997778579) \\&+ \overline{t_k}\cdot  (-0.095675525034475) + \overline{t_{k-2}}\cdot  0.76763340402437 + \overline{t_{k-3}}\cdot  -0.39440568040737\\
y&=y-0.0060646128345046 \cdot  \cos(s) + 0.0020001498578622 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.33592101311812 + \overline{v_{k-1}}\cdot  0.29300517495018 + \overline{v_{k-2}}\cdot  (-0.081008193206153) + \overline{v_{k-3}}\cdot  0.65788763344974 \\&+ \overline{n_k}\cdot  0.49443884363231 + \overline{n_{k-1}}\cdot  (-0.14546781842356) + \overline{n_{k-3}}\cdot  2.6823414085615 \\&+ \overline{t_k}\cdot  0.1016495953276 + \overline{t_{k-2}}\cdot  0.71731959720856 + \overline{t_{k-3}}\cdot  0.49214444361749\\
y&=y+ 0.049365601607823 \cdot  \cos(s) -0.11114151241192 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.12313743363216) + \overline{v_{k-1}}\cdot  0.054325631391886 + \overline{v_{k-2}}\cdot  0.44523931940928 + \overline{v_{k-3}}\cdot  (-0.29965463301554) \\&+ \overline{n_k}\cdot  (-6.5319333906529) + \overline{n_{k-1}}\cdot  0.10456259009935 + \overline{n_{k-3}}\cdot  6.4734343584366 \\&+ \overline{t_k}\cdot  (-0.26451209770456) + \overline{t_{k-2}}\cdot  (-0.032280273582581) + \overline{t_{k-3}}\cdot  0.10726501135496\\
y&=y-0.18747373692342 \cdot  \cos(s) + 0.13584975865301 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.4006160658585 + \overline{v_{k-1}}\cdot  (-0.21283083202285) + \overline{v_{k-2}}\cdot  (-0.068312192457003) + \overline{v_{k-3}}\cdot  0.20618038289807 \\&+ \overline{n_k}\cdot  (-2.4511429858129) + \overline{n_{k-1}}\cdot  (-0.052334463687479) + \overline{n_{k-3}}\cdot  2.1087287187327 \\&+ \overline{t_k}\cdot  (-0.29875820186742) + \overline{t_{k-2}}\cdot  (-0.80743257379728) + \overline{t_{k-3}}\cdot  -0.66277883318956\\
y&=y+ 0.10927821611682 \cdot  \cos(s) -0.16980173269916 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.27701655987021 + \overline{v_{k-1}}\cdot  (-0.18859547586744) + \overline{v_{k-2}}\cdot  0.58822055078665 + \overline{v_{k-3}}\cdot  (-0.169705037502) \\&+ \overline{n_k}\cdot  (-5.2430251385084) + \overline{n_{k-1}}\cdot  (-0.014934294393431) + \overline{n_{k-3}}\cdot  4.9214388242382 \\&+ \overline{t_k}\cdot  (-1.0679270513252) + \overline{t_{k-2}}\cdot  (-0.82405027514928) + \overline{t_{k-3}}\cdot  0.48310683714814\\
y&=y+ 0.036114118023937 \cdot  \cos(s) + 0.02177097381349 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.76812303505839 + \overline{v_{k-1}}\cdot  (-0.21549786958375) + \overline{v_{k-2}}\cdot  (-0.56064503290645) + \overline{v_{k-3}}\cdot  0.41941568588048 \\&+ \overline{n_k}\cdot  (-4.3821816595169) + \overline{n_{k-1}}\cdot  (-0.28204286471956) + \overline{n_{k-3}}\cdot  0.95775292601595 \\&+ \overline{t_k}\cdot  0.27313504970283 + \overline{t_{k-2}}\cdot  (-0.60848484526538) + \overline{t_{k-3}}\cdot  -1.019094971232\\
y&=y+ 0.022156600266183 \cdot  \cos(s) -0.071298248083971 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.98327764735988 + \overline{v_{k-1}}\cdot  0.076033702180366 + \overline{v_{k-2}}\cdot  (-0.15793453359845) + \overline{v_{k-3}}\cdot  0.71950716503931 \\&+ \overline{n_k}\cdot  (-0.47354412535569) + \overline{n_{k-1}}\cdot  0.24127484360951 + \overline{n_{k-3}}\cdot  0.7659084120113 \\&+ \overline{t_k}\cdot  0.83200048015893 + \overline{t_{k-2}}\cdot  (-0.26268813906648) + \overline{t_{k-3}}\cdot  -1.2054529288702\\
y&=y-0.064087023224232 \cdot  \cos(s) + 0.079679907842945 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.8916980909554 + \overline{v_{k-1}}\cdot  0.076720953877967 + \overline{v_{k-2}}\cdot  (-0.45019093325921) + \overline{v_{k-3}}\cdot  0.36548965680332 \\&+ \overline{n_k}\cdot  0.21068731421127 + \overline{n_{k-1}}\cdot  0.44490303586371 + \overline{n_{k-3}}\cdot  0.56120813986296 \\&+ \overline{t_k}\cdot  (-0.011496002719082) + \overline{t_{k-2}}\cdot  (-1.0078130376216) + \overline{t_{k-3}}\cdot  0.44490598310089\\
y&=y-0.15284350315258 \cdot  \cos(s) -0.070732848135661 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  (-0.29091377550497) + \overline{v_{k-1}}\cdot  (-0.32912199707093) + \overline{v_{k-2}}\cdot  (-0.4298165020062) + \overline{v_{k-3}}\cdot  0.59574409894471 \\&+ \overline{n_k}\cdot  (-2.8754419446166) + \overline{n_{k-1}}\cdot  (-0.086683256116449) + \overline{n_{k-3}}\cdot  1.6454379877669 \\&+ \overline{t_k}\cdot  0.25938854556758 + \overline{t_{k-2}}\cdot  (-0.97475354208878) + \overline{t_{k-3}}\cdot  0.10806979810943\\
y&=y-0.24232239333697 \cdot  \cos(s) -0.081581950149644 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.17893792251594 + \overline{v_{k-1}}\cdot  0.16954010216603 + \overline{v_{k-2}}\cdot  0.51201140789787 + \overline{v_{k-3}}\cdot  (-1.0523064897854) \\&+ \overline{n_k}\cdot  (-1.167040024119) + \overline{n_{k-1}}\cdot  (-0.10698900931449) + \overline{n_{k-3}}\cdot  1.626316962915 \\&+ \overline{t_k}\cdot  (-0.0072453457587813) + \overline{t_{k-2}}\cdot  (-0.0092316317016417) + \overline{t_{k-3}}\cdot  -0.011162384381199\\
y&=y-0.29386565010948 \cdot  \cos(s) + 0.65135209764835 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.0063606034904218 + \overline{v_{k-1}}\cdot  0.071501324376628 + \overline{v_{k-2}}\cdot  (-0.48863571434896) + \overline{v_{k-3}}\cdot  0.14135732422574 \\&+ \overline{n_k}\cdot  (-3.4840052050416) + \overline{n_{k-1}}\cdot  0.24675318111587 + \overline{n_{k-3}}\cdot  (-1.6449484288109) \\&+ \overline{t_k}\cdot  (-0.56672822756487) + \overline{t_{k-2}}\cdot  (-1.1501495313197) + \overline{t_{k-3}}\cdot  -0.86455150613677\\
y&=y+ 0.055390731056039 \cdot  \cos(s) + 0.04739767721103 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  1.0269981893507 + \overline{v_{k-1}}\cdot  (-0.082951886079255) + \overline{v_{k-2}}\cdot  0.22708479699296 + \overline{v_{k-3}}\cdot  (-0.42111959856584) \\&+ \overline{n_k}\cdot  0.27806038339095 + \overline{n_{k-1}}\cdot  (-0.11855285965653) + \overline{n_{k-3}}\cdot  (-0.80495473348493) \\&+ \overline{t_k}\cdot  1.1492815648505 + \overline{t_{k-2}}\cdot  0.15774188423831 + \overline{t_{k-3}}\cdot  -0.28823932935373\\
y&=y-0.42557167422989 \cdot  \cos(s) + 0.13885157000596 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.39965111097893 + \overline{v_{k-1}}\cdot  0.13710749855091 + \overline{v_{k-2}}\cdot  0.26849312382293 + \overline{v_{k-3}}\cdot  0.81666717467468 \\&+ \overline{n_k}\cdot  2.4997475602785 + \overline{n_{k-1}}\cdot  0.25828347457931 + \overline{n_{k-3}}\cdot  5.392425989108 \\&+ \overline{t_k}\cdot  0.3779202358358 + \overline{t_{k-2}}\cdot  (-1.3746223919269) + \overline{t_{k-3}}\cdot  1.2911730603403\\
y&=y+ 0.025005799753262 \cdot  \cos(s) + 0.02534531537369 \cdot  \sin(s)\\
s&=\overline{v_k}\cdot  0.75270340663039 + \overline{v_{k-1}}\cdot  0.2712441950529 + \overline{v_{k-2}}\cdot  (-0.051316020031209) + \overline{v_{k-3}}\cdot  0.81218286537538 \\&+ \overline{n_k}\cdot  0.33584399773702 + \overline{n_{k-1}}\cdot  (-0.073024756911837) + \overline{n_{k-3}}\cdot  (-0.078118634689732) \\&+ \overline{t_k}\cdot  (-1.044296955018) + \overline{t_{k-2}}\cdot  (-0.78289236871129) + \overline{t_{k-3}}\cdot  1.2210752978127\\
y&=y+ 0.0032374278105146 \cdot  \cos(s) + 0.068392460514006 \cdot  \sin(s)\\
&\textbf{Unscaling and return}\\
& \operatorname{return }\ y \cdot  5.1489801806121 + 14.886169231027
\end{align*}