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DRAFT VERSION MAY 20, 2021
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Estimating Magnetic Filling Factors From Simultaneous Spectroscopy and Photometry: Disentangling Spots, Plage, and
Network
T. W. MILBOURNE ,1, 2 , ∗D. F. PHILLIPS ,2N. LANGELLIER ,1, 2 A. MORTI ER ,3 , 4 R. D. HAYWOOD ,2 , 5, †S. H. SA AR ,2
H. M. CEG LA ,6 , 7, ‡A. COLLIER CAMERON ,8X. DUMUSQUE ,6D. W. LATHAM ,2L. MALAVOLTA ,9J. MALD ONA DO ,10
S. THOM PS ON ,3A. VANDERBURG,11 C. A. WATSON ,12 L. A. BU CHHAVE ,13 M. CECCO NI ,14 R. COSENTINO ,1 4
A. GHED INA ,14 M. GONZALEZ,14 M. LODI ,14 M. LÓPEZ-MORALES ,2A. SOZZETT I ,15 AN D R. L. WALSWORTH 16, 17, 18
1Department of Physics, Harvard University, Cambridge MA 02138, USA
2Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA
3Astrophysics Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK
4Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
5Astrophysics Group, University of Exeter, Exeter EX4 2QL, UK
6Observatoire de Genève, Université de Genève, 51 chemin des Maillettes, 1290 Versoix, Switzerland
7Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
8Centre for Exoplanet Science, SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK
9Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Vicolo dell‘Osservatorio 3, I-35122 Padova, Italy
10INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy
11Department of Astronomy, University of Wisconsin, Madison, WI 53706-1507, USA
12Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Belfast, UK
13DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 328, DK-2800 Kgs. Lyngby, Denmark
14INAF-Fundacion Galileo Galilei, Rambla Jose Ana Fernandez Perez 7, E-38712 Brena Baja, Spain
15INAF-Osservatorio Astrofisico di Torino, via Osservatorio 20, 10025 Pino Torinese, Italy
16Department of Physics, University of Maryland, College Park, MD 20742, USA
17Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA
18Quantum Technology Center, University of Maryland, College Park, MD 20742, USA
ABSTRACT
State of the art radial velocity (RV) exoplanet searches are limited by the effects of stellar magnetic activ-
ity. Magnetically active spots, plage, and network regions each have different impacts on the observed spectral
lines, and therefore on the apparent stellar RV. Differentiating the relative coverage, or filling factors, of these
active regions is thus necessary to differentiate between activity-driven RV signatures and Doppler shifts due
to planetary orbits. In this work, we develop a technique to estimate feature-specific magnetic filling factors
on stellar targets using only spectroscopic and photometric observations. We demonstrate linear and neural
network implementations of our technique using observations from the solar telescope at HARPS-N, the HK
Project at the Mt. Wilson Observatory, and the Total Irradiance Monitor onboard SORCE. We then compare
the results of each technique to direct observations by the Solar Dynamics Observatory (SDO). Both imple-
mentations yield filling factor estimates that are highly correlated with the observed values. Modeling the solar
RVs using these filling factors reproduces the expected contributions of the suppression of convective blueshift
and rotational imbalance due to brightness inhomogeneities. Both implementations of this technique reduce the
overall activity-driven RMS RVs from 1.64 m s−1to 1.02 m s−1, corresponding to a 1.28 m s−1reduction in the
RMS variation. The technique provides an additional 0.41 m s−1reduction in the RMS variation compared to
traditional activity indicators.
Keywords: exoplanets — techniques: radial velocities — Sun: activity — planets and satellites: detection
∗Corresponding Author: tmilbourne@g.harvard.edu
†NASA Sagan Fellow
‡CHEOPS Fellow, SNSF NCCR-PlanetS
UKRI Future Leaders Fellow
1. INTRODUCTION
State of the art radial velocity (RV) searches for low-mass,
long-period exoplanets are limited by the effects of stellar
magnetic activity. An Earth-mass planet in the habitable zone
of a Sun-like star has an RV amplitude on the order of 10
arXiv:2105.09113v1 [astro-ph.SR] 19 May 2021
2 T. MILBO URNE ET AL.
Figure 1. A representative HMI map of the three classes of active regions considered in this work. Spots, plage, and network are identified
using the thresholding algorithm described by Haywood et al. (2016) and MH19, with the threshold values given by Yeo et al. (2013). This
algorithm is briefly recapped in Sec. 2.3. Image taken January 1st, 2015 at 0:0:0.00 UT.
cm s−1. However, stellar activity processes on host stars,
such as acoustic oscillations, magnetoconvection, suppres-
sion of convective blueshift, and long-term activity cycles,
can produce signals with amplitudes exceeding 1 m s−1. A
variety of techniques exist to mitigate the effect of these pro-
cesses on the measured RVs: Chaplin et al. (2019) discuss
optimal exposure times to average out acoustic oscillations;
Cegla (2019) and Meunier et al. (2017) present strategies
for mitigaing the effects of granulation; and Aigrain et al.
(2012), Rajpaul et al. (2015), Langellier et al. (2020), Hay-
wood et al. (2020), and numerous others discuss statistically
and physically-driven techniques for removing the effects of
large-scale magnetic regions from RV measurements.
On timescales of the stellar rotation period, the apparent
radial velocity is modulated by three main types of active re-
gions: dark sunspots; large, bright plage; and small, bright
network regions. These different regions may be identified
using full-disk solar images, as shown in Fig. 1.Milbourne
et al. (2019) (hereafter referred to as MH19) found that the
large-scale photospheric plage contribute differently to the
solar suppression of convective blueshift than the smaller
network. Failure to account for this different contribution
leads to a significant RV shift over the 800 day span their
of observations. Some of the long-term variation reported in
MH19 may also be attributed to instrumental systematics: re-
reducing the HARPS-N solar data with the ESPRESSO DRS
(Pepe, F. et al. 2021;Dumusque et al. 2020) reduces this shift
from 2.6 ±0.3 m s−1to 1.6 ±0.5 m s−1. However, the re-
maining RV shift can only be fully removed by properly ac-
counting for network regions in the calculated activity-driven
RVs. While this analysis is possible on the Sun using high-
resolution full disk images, traditional spectroscopic activity
indicators, such as the Mt. Wilson S-Index (Wilson 1968;
Linksy & Avrett 1970) and the derivative index log(R0
HK )
(Vaughan et al. 1978;Noyes et al. 1984) do not differentiate
between large and small active regions. A new activity in-
dex or combination of activity indices is therefore necessary
to successfully model the suppression of convective blueshift
on stellar targets.
In this work, we demonstrate a new technique using simul-
taneous spectroscopy and photometry to estimate spot, plage,
and network filling factors, and demonstrate that these filling
factors may be used to model RV variations. In Section 2,
we discuss the solar data used by our technique. An analyt-
ical implementation of the technique is described in Section
3, and a neural network implementation is presented in Sec-
ESTIMATING MAGNET IC FILLING FAC TORS FROM SIMULTANEOUS SPECTROSCOPY AND PHOTOME TRY 3
tion 4. The resulting solar filling factors, a model of the solar
RVs, and possible applications to stellar targets are analyzed
in Section 5.
2. MEASUREMENTS
2.1. HARPS-N/Mt. Wilson Survey
We use the HARPS-N solar telescope (Dumusque et al.
2015;Phillips et al. 2016) measurements of the S-index,
as described in MH19 and Collier Cameron et al. (2019).
The S-index quantitatively represents activity-driven chro-
mospheric re-emission in the Calcium II H and K lines. The
presence of spots, plage, and network all increase the S-
index. The solar telescope takes exposures every five min-
utes while the Sun is visible. Each measurement of the S-
index has an average precision of 2.5×10−4, or a fractional
uncertainty of 0.0016.
Note that HARPS-N solar telescope observations began in
mid 2015. To cover the rest of the solar cycle, we use data
from the Mt. Wilson S-index survey, as presented by Ege-
land et al. (2017). Observations from the two instruments
overlap between July 2015 and February 2016 (JD 2457222
and JD 2457444), allowing us to combine these time series.
The solar telescope dataset is rescaled so that the points in
the overlapping time interval have the same mean and vari-
ance as the Mt. Wilson data from the same time interval, as
described in Haywood et al. (2020). The resulting combined
dataset is shown in the top panel of Fig. 2.
2.1.1. HARPS-N Solar RVs
We use the HARPS-N solar telescope’s measurements of
the solar RVs to assess our ability to model realistic RV vari-
ations using our estimated filling factors. 1(Phillips et al.
2016;Collier Cameron et al. 2019;Dumusque et al. 2020),
as well as our estimated values derived from the linear and
MLP techniques. The HARPS-N RVs used in this work span
the period from July 2015 to October 2017, with exposures
taken every five minutes while the Sun is visible. Each RV
measurement has an average precision of 23 cm s−1.
2.2. SORCE
We use the Total Irradiance Monitor (TIM) onboard the
Solar Radiation and Climate Experiment (SORCE) (Rottman
2005;Kopp & Lawrence 2005;Kopp et al. 2005) to measure
photometry for the whole solar cycle. The total solar irradi-
ance (TSI) is the solar analogue of the light curves obtained
by Kepler, K2, TESS, and CHEOPS (Borucki et al. 2010;
Howell et al. 2014;Ricker et al. 2014;Cessa et al. 2017),
1We use publicly available HARPS-N solar telescope observations reduced
using the most recent ESPRESSO pipeline, available at https://dace.unige.
ch/sun/
though the Sun’s proximity means it can be observed con-
tinuously over much longer periods. The TIM level 3 data
products are averaged over 6 hours, with a precision of 0.005
W/m2.
We expect the overall brightness of the Sun to vary with the
stellar cycle. Its relative brightness increases with the pres-
ence of plage and network, and decreases with the presence
of spots. This modulation makes the TSI, and stellar light
curves in general, useful tools for isolating the effects of stel-
lar magnetic activity (Aigrain et al. 2012). The time series of
the TSI is shown in the second panel of Fig. 2.
2.3. SDO
We use images from the Helioseismic and Magnetic Im-
ager (HMI) instrument onboard the Solar Dynamics Obser-
vatory (SDO, Schou et al. 2012;Pesnell et al. 2012;Couvidat
et al. 2016) to independently calculate solar filling factors.
HMI measures the 6173.3 Å iron line at six points in wave-
length space using two polarizations. From these measure-
ments, they reconstruct the Doppler shift and magnetic field
strength along with the continuum intensity, line width, and
line depth at each point on the solar disk.
Spots, plage, and network are identified on HMI images
using a simple threshold algorithm:
• An HMI pixel is considered magnetically active if the
radial component of the magnetic field is over three
times greater than the expected noise floor: |Br|>
3σBr.
• Active pixels below the intensity threshold of Yeo et al.
(2013), such that I<0.89Iquiet , are labelled as spots.
Here, Iquiet is the average intensity of inactive pixels on
a given image.
• Active regions exceeding the above intensity thresh-
old that span an area > 20 micro-hemispheres (that
is, 20 parts per million of the visible hemisphere), or
60 Mm2, are labelled as plage.
• Active regions exceeding the intensity threshold that
span an area < 20 micro-hemispheres are labelled as
network.
These calculations are explained in further detail in MH19.
The resulting filling factors for each feature are plotted in
the bottom three panels of Fig. 2. We use one HMI image
taken every four hours in our analysis. The photon noise at
disk center for the magnetograms and continuum intensity
for these HMI images are σBr= 8 G and σIc= 0.01% respec-
tively (Couvidat et al. 2016). This corresponds to uncertain-
ties <0.1% in the resulting magnetic filling factors.
Since SDO/HMI allows us to perform precise direct, inde-
pendent measurements of the three filling factors of interest,
we use these results as the "ground truth" in our analysis.
4 T. MILBO URNE ET AL.
Figure 2. Time series of solar observations used in this work. From top to bottom: Mt. Wilson and HARPS-N solar telescope observations
of the calcium S-index (red); Total Solar Irradiance (TSI) from SORCE/TIM (blue); and SDO/HMI plage, spot, and network filling factors
(black). Note that the consistent overall shapes of the S-index, TSI, and bright filling factors, and that dips in the TSI are coincident with peaks
in the spot filling factor. Observations are taken between April 2010 through October 2017. Note the two different reds used in the S-index
plot: the darker red points correspond to measurements by Mt. Wilson, and bright red points are from the HARPS-N solar telescope. Note that
due to an instrumental anomaly, no TSI data is availble from SORCE/TIM from mid July 2013 until March 2014 - we therefore do not use any
times in this period in our analysis (Kopp & Lawrence 2005;Kopp et al. 2005)
.
ESTIMATING MAGNET IC FILLING FAC TORS FROM SIMULTANEOUS SPECTROSCOPY AND PHOTOME TRY 5
An SDO analogue does not exist for non-solar stars, so if
we wish to determine feature-specific filling factors of stellar
targets we must make indirect estimates of the filling fac-
tors using spectroscopic and photometric data. In the next
section, we discuss two processes to do so. To mitigate the
effects of acoustic oscillations, granulation, and other short-
timescale activity process, we take daily averages of each set
of observations used in our analysis. We also interpolate the
HARPS-N/Mt. Wilson observations, SORCE/TIM observa-
tions, and SDO/HMI filling factors onto a common time grid
of one observation each day, when all three instruments have
measurements.
3. LINEAR TECHNIQUE
3.1. Modelling Irradiance Variations Using Filling Factors
In MH19, the authors reproduce the observed TSI using a
linear combination of the spot and plage filling factors. Fol-
lowing Meunier et al. (2010), they use the SDO/HMI derived
plage and spot filling factors, and assume that the solar irra-
diance follows the Stefan-Boltzmann law for blackbodies:
T SI =Aσ(1 −fs pot −fbright)T4
quiet
+fspot (Tquiet +∆Ts pot )4
+fbright (Tquiet +∆Tbright)4(1)
where σis the Stefan-Boltzmann constant, A= (R/1 AU)2
is a geometrical constant relating the energy emitted at the so-
lar surface to the energy received at Earth, Tquiet is the quiet
Sun temperature, ∆Tspot and ∆Tbright are the effective tem-
perature contrasts of spots and plage/network regions, and
fspot and fbright are the HMI spot and plage/network filling
factors. Expanding as a power series yields the following
approximation:
T SI ≈ AσT4
quiet 1+4∆Tspot
Tquiet
fspot +4∆Tbright
Tquiet
fbright (2)
In MH19, the authors show that HMI observations of fill-
ing factors may be used to reproduce SORCE TSI given tem-
perature contrasts for plage/network features and spots and
the effective temperature of the quiet Sun. In this work, we
invert the process and use the resulting effective temperatures
of each type of active region, along with the correlations be-
tween the TSI, S index, and filling factors demonstrated in
MH19, to reproduce the observed magnetic filling factors for
each type of active region. The potential stellar applications
of this technique are discussed in more detail in Sec. 5.2.
We begin by fitting the SORCE TSI to Eq. 2using
the SDO/HMI measurements of fs pot and fbright. This fit
yields Tquiet = 5769.85 ±0.01 K, ∆Tspot =−525 ±8 K, and
∆Tspot = 46.1±0.5 K. Also note that the solar radius varies as
Figure 3. A plot of the SORCE TSI versus the HARPS-N/Mt. Wil-
son S index. The color of each point corresponds to the value of
fspot . We see that the S index is highly correlated with the TSI, as
expected. We may use this correlation to estimate the plage/network
filling factors on the Sun. However, increased spot coverage results
in a lower TSI value for a given S index, which will bias our esti-
mate of bright region filling factor. This, in turn, will result in a less
accurate estimate of the spot filling factor. The black line shows the
result of the straightforward linear fit of TSI and S index, which is
biased as described above. To isolate the plage and network driven
TSI variations, we find the 50% most densely clustered points in
the above scatter plot, and fit a line to the upper boundary of this
region. (This choice in point density is arbitrary, but the resulting
best-fit line is robust to variations in this parameter.) The resulting
fit line, shown in red, is unbiased by the presence of spots.
a function of wavelength. To be consistent with HMI, we use
R= 695982±13 km, the solar radius measured at 6173.3 Å
(Rozelot et al. 2015).
3.2. Differentiating Bright and Dark Regions
The brightness of a Sun-like star may be modulated by the
long-term stellar activity cycle (e.g., the 11-year solar cycle).
Since ∆Tbright >0 and ∆Tspot <0, we see from Eq. 2that
TSI variations (on timescales of the rotation period) below
the current value from the activity cycle must be the result of
spots. Since the Sun is plage dominated (Shapiro et al. 2016),
plage and network are the primary source of variation of the
TSI and S-index, with spots making negligible contributions
to the variability of the irradiance on timescales of the solar
cycle. This is also visible in comparing the plots of the TSI, S
index, fplage, and fnt wk shown in Fig. 2, and is also discussed
in detail by MH19. We may therefore use a linear transfor-
mation of the S-index to provide an initial estimate of fbright ,
6 T. MILBO URNE ET AL.
and then use the TSI to estimate fs pot. The full calculation is
as follows:
(1) We begin by assuming that the S-index is directly pro-
portional to the total plage and network filling factor, as
shown in Fig. 3(that is, fbright,1=m1SH K +b1), and that
the plage and network are the dominant drivers of TSI vari-
ation. Our first estimate of the spot filling factor is therefore
fspot ,1= 0. We then estimate the values of m1and b1by fitting
the TSI as a linear transformation of the S-index:
T SI fit ,1(m1,b1) = AσT4
quiet 1+4∆Tbright
Tquiet
(m1SHK +b1).
(3)
Note that we have included the physical constants for normal-
ization though they are degenerate with the fit parameters.
It is not sufficient to perform a simple linear fit to the TSI
and S index. In the above step, we model the activity-driven
variations of the TSI due to the presence of bright regions.
However, as shown in Fig. 3, the presence of spots produces
scatter in this relationship. This scatter is in one direction: as
fspot increases, the observed TSI for a given value of the S-
index decreases. To isolate the activity-driven TSI variations
due to fbr ight, we determine the 50% most densely clustered
points in Fig. 3, and fit the upper boundary of this region.
The best fit line of this upper bound gives us the values of m1
and b1used above.
(2) Next, we assume any deviation from the fit above are
driven by spots, which are not included in this model. We
then make a second estimate of the spot filling factor, fspot,2,
from the residuals to the above fit:
fspot ,2=
T SI −T S If it,1(m1,b1)
4Aσ∆TspotT3
quiet
,T SI −T S If it,1≤0
0,T SI −T S If it,1>0.
(4)
Essentially, any point below the line of best fit in Fig. 3is as-
sumed to be due to spot-driven brightness variations. This in-
creases the importance of avoiding spot-driven biases in Step
1. If a simple linear fit is used in Step 1 instead of the fit
to the upper boundary described above, the presence of spots
will reduce the slope of the best-fit line, which will result in
an artificially reduced fbright value, and will also exclude real
spot-driven variations from our calculation of fs pot,2.
(3) We determine our final estimate of fbright and fspot by
fitting the following expression to the TSI:
T SI fit ,2=AσT4
quiet 1+4∆Tspot
Tquiet a2fspot ,2
+4∆Tbright
Tquiet
(m2SHK +b2).(5)
where our estimated values of fbright and fspot are given by
fbright =m2SHK +b2(6)
m12.02 ±0.07
b1−0.31 ±0.01
a20.9912 ±0.0008
m22.072 ±0.002
b2−0.3169 ±0.0003
Table 1. Best-fit parameters for the linear filling factor estimation
technique. As expected, m1and m2are consistent within error bars,
as are b1and b2. Similarly, a2is very close to 1, as expected..
fspot =a2fs pot,2(7)
and the parameters a2,m2, and b2are determined by the
above fit. The resulting best-fit parameters derived from the
solar case are given in Table 1. Note that we do not expect
m2and b2to be very different from the parameters m1and
b1found previously, nor do we expect a2to be very different
from 1. However, since we exclude any negative residuals
from our estimate of fspot in Step 2 above, we perform this
final fit in case this excluded information changes the best-fit
parameters in any way.
Note that this technique only requires knowledge of the
star’s distance and radius, along with estimates of the spot
and plage/network temperature contrasts and the quiet star
effective temperature - this means that it can be used to es-
timate filling factors without prior knowledge of the filling
factors from full-disk images. If the plage and network fea-
tures are only being used to decorrelate activity-driven RV
variations, only time series correlated with the spot, network,
and plage filling factors are needed, and the above terms may
be absorbed into the fit coefficients in Eqs. 3,5—this is dis-
cussed further in Sec. 5.2.
3.3. Differentiating the Network and Plage Filling Factor
In the discussion above, we extract fbright, the combined
plage and network filling factor. However, we may consider
these two separately by adding a network term to Eq. 1:
T SI =Aσ(1 −fs pot −fplage −fntwk )T4
quiet
+fspot (Tquiet +∆Ts pot )4
+fplage(Tquiet +∆Tplage)4
+fntwk (Tquiet +∆Tntwk )4.(8)
Fitting this equation to the TSI using the SDO observed
filling factors reveals that the plage and network have dis-
tinct effective temperatures, ∆Tplage = 32 ±1Kand ∆Tntwk =
79±2K. This is consistent with the intensity maps produced
by HMI, which show that network regions are, on average,
indeed brighter than plage These temperature contrasts are
ESTIMATING MAGNET IC FILLING FAC TORS FROM SIMULTANEOUS SPECTROSCOPY AND PHOTOME TRY 7
necessary for seperating the plage and network contributions
to the filling factor.
Setting Eq. 1equal to Eq. 8and expanding as a power se-
ries, we find
fbright ∆Tbright ≈fplage ∆Tplage +fntwk ∆Tntwk .
Since areas are additive, we also expect
fbright =fplage +fntwk .
Combining these equations and solving for fntwk and fplage in
terms of fbright yields the following expressions:
fntwk =∆Tplage −∆Tbright
∆Tplage −∆Tntwk
fbright +B,(9)
and
fplage =∆Tbright −∆Tnt wk
∆Tplage −∆Tntwk
fbright −B.(10)
Note the prefactors for each estimate, which simply rescale
the brightness contributions of each class of active region to
account for the different effective temperatures. Also note
the offset Bin our estimate of fplage: this accounts for the
fact that fplage goes to 0 at solar minimum, while fntwk has
a basal value at solar minimum. In this analysis, the value
of Bmay be found from the expected value of fbright at solar
minimum:
B=∆Tbright −∆Tntwk
∆Tplage −∆Tntwk
minfbright .(11)
Determining this offset therefore requires TSI and S index
observations taken at solar minimum, which may increase
the observational load associated with this technique. How-
ever, modeling the effects of network on the activity-driven
solar RV variations only requires a quantity correlated with
fplage. The value of this offset is therefore unimportant for
our purposes.
4. MACHINE LEARNING TECHNIQUE
While machine learning techniques are predicatively pow-
erful, their black-box nature makes them not physically ex-
planatory, and therefore not necessarily useful for some sci-
entific applications. However, the existence of a clear causal
connection between the S-index, TSI, and filling factors
makes machine learning a strong candidate for the prob-
lem of estimating feature-specific magnetic filling factors
from spectroscopic and photometric information. We already
know the physics connecting these variables, and can there-
fore have machine learning "discover" and refine the rela-
tionships found above. A neural network used as a universal
function approximator (Cybenko 1989) may be able to de-
termine subtle details of these relationships that are not in-
corporated into our linear model, such as the different effects
Hidden Layer Sizes α β
(64,64) 0.0001 0.001
Table 2. Hyperparameter values for MLP filling factor calculation
as optimized from cross-validation. Here αgives the L2regulariza-
tion parameter and βis the learning rate.
of network vs. plage, how underlying spatial distributions of
active regions affect the resulting filling factors and activity
indicators, and the correlations between spots and plage.
We therefore compare the linear technique discussed in the
previous section with a type of neural network known as a
Multilayer Perceptron (MLP, Hinton 1989). The MLP con-
sists of an input layer, several fully-connected hidden layers,
and an output layer. It is one of the simplest neural networks
that may be used as a universal function approximator, mak-
ing it ideal for this application. We implement the MLP using
the MLPRegressor class in the scikit-learn package
in Python (van Rossum 1995;Pedregosa et al. 2011).
We train the MLP using the TSI and S-index inputs, and
using the SDO plage, spot, and network filling factors as out-
puts. 75% of the total available data (taken over the whole
solar cycle) is used for training, with 25% set aside to test the
performance of the trained network. The MLP uses two hid-
den network layers, each with 64 neurons. We optimized the
size and number of these layers as well as the L2regulariza-
tion parameter, α, which combats overfitting by constraining
the size of the fit parameters as measured with an L2norm;
and the learning rate, β, which controls the step-size in the
parameter space search using five-fold cross validation. That
is, we randomly shuffled the training data and divided it into
five groups. We trained the network on four of these groups,
and then tested the network on the remaining group. We re-
peated this process using each of the five groups as a test set
to mitigate the effects of overfitting on our network, and then
repeated the entire five-fold process using each combination
of network parameters to determine which combination of
hyperparameters resulted in the best performance. The re-
sulting values are summarized in Table 2. The network was
optimized to minimize square error using a stochastic gradi-
ent descent algorithm (SGD), and was trained for a maximum
of 104steps (though the algorithm may stop training earlier
once the network converges.)
Note that we may also use this MLP approach to fit the
solar RVs directly using the TSI and S-index, without first
computing magnetic filling factors. In Sec. 5.1, we compare
a direct MLP fit of this form to RV models dervived from
our estimated filling factors to determine if there is any ad-
ditional RV information in the TSI and S index which is not
incorporated into our filling factor estimates.
8 T. MILBO URNE ET AL.
5. RESULTS AND DISCUSSION
Fig. 4shows that both the linear and MLP-based tech-
niques successfully reproduce the directly-observed values
of fspot ,fplage , and fntwk . Fig. 5shows the same information
as Fig. 4, but for three 230 day regions taken in the middle
of the solar cycle, at solar maximum, and at solar minimum.
We see that, again, both the linear and MLP techniques are
able to reproduce the SDO-measured values of fs pot ,fplage,
and fntwk at all points in the stellar activity cycle on these
timescales.
Note that there is a systematic ∼0.004 offset between the
linear estimates of fntwk and the SDO measured values. This
is likely the result of the significant covariance between b1
and AσT4
quiet in Eq. 3. Any systematic errors in the mea-
sured values of Rand T4
quiet will change the resulting value
of b1, resulting in an offset in the estimated values of fntwk .
Small changes to these parameters can dramatically change
the observed offset in fntwk : artificially increasing T4
quiet by
0.15 K eliminates the offset entirely. This is well below the
precision achieved for measurement of stellar temperatures.
While we attain good precision in the solar case, in general
linear estimates of fntwk should assumed to be true up to a
constant offset. As stated previously, using these filling fac-
tors to remove activity-driven signals from RV measurements
only requires values correlated with the filling factor value,
making this offset unimportant.
We also note that, while Rand T4
quiet are assumed to be
constants in our model, they do change in time as the result
of physical processes not included in our model. These quan-
tities also vary with wavelength: Since here we are using the
Ca II H&K lines and integrated visible intensity to reproduce
filling factors measured at 6173.3 Å, uncertainties in these
parameters associated with their wavelength dependence are
inevitable. Indeed, Meunier et al. (2010) note that measured
filling factors will vary by 20% to 50% as a result of these de-
pendencies and other definitional differences: our estimated
fntwk values are certainly consistent with the SDO measured
values within these margins.
In Table 3, we list the Pearson correlation coefficients be-
tween the HMI derived filling factors and our estimates from
the linear and MLP techniques. (For the sake of consistency,
note that for both the linear and MLP estimates, we compute
correlation coefficients only for results generated using the
fraction of data reserved for testing the MLP.) We see that
both techniques reproduce the information contained in the
HMI filling factors, with the MLP performing slightly better
than the linear model on all three filling factors. This may in-
dicate that there is some additional information about the fill-
ing factors present in the TSI and S index observations that
is not being used by the linear technique. However, given
the high degree of correlation produced by both techniques,
Spots Plage Network
Linear 0.81 0.87 0.81
MLP 0.85 0.87 0.89
Table 3. Pearson correlation coefficients between HMI ground-
truth filling factors and the linear and MLP estimates for each class
of filling factor.
Filling Factor Source r∆vconv,RV (fspot ,fntwk ,fplage )
HMI 0.92
Linear Estimate 0.84
MLP Estimate 0.83
Table 4. Pearson correlation coefficients between HMI derived es-
timate of the suppression of convective blueshift, ∆vconv and the
activity driven RVs derived from Eq. 13. The very high correla-
tion coefficients indicate that the plage and network filling factors
successfully estimate the RV contribution of the suppression of con-
vective blueshift, as expected from MH19.
Filling Factor Source r∆vphot ,RV(fs pot )
HMI 0.62
Linear Estimate 0.70
MLP Estimate 0.67
Table 5. Pearson correlation coefficients between HMI derived esti-
mate of the photometric velocity shift, ∆vphot and the activity driven
RVs derived from Eq. 14. The relatively high correlation coeffi-
cients indicate that the spot filling factors successfully estimate the
photometric RV shifts, as expected from MH19.
we can use both estimates of the magnetic filling factors to
reduce the effects of activity on observed RVs.
5.1. Application to Solar RVs
In MH19, the authors found that the HARPS-N solar radial
velocities were well-represented by a linear combination of
∆vconv, the suppression of convective blueshift, and ∆vphot,
the photometric velocity shift due to bright and dark active
regions breaking the symmetry of the solar rotational profile:
RV =A1∆vphot +B1∆vconv +RV0.(12)
Here, we see if we can perform a similar reconstruction us-
ing our estimates of the magnetic filling factors. Since the
presence of active regions drives the suppression of convec-
tive blueshift, we expect the ∆vconv to be proportional to the
spot, plage, and network filling factors. Based on the re-
sults of MH19, we also expect network and plage regions
to have different contributions to ∆vconv. We therefore model
the suppression of convective blueshift as:
∆vconv =B fspot +C f plage +D fnetwork +E.(13)
ESTIMATING MAGNET IC FILLING FAC TORS FROM SIMULTANEOUS SPECTROSCOPY AND PHOTOME TRY 9
Figure 4. Comparison of the SDO/HMI-measured magnetic filling factors (black) to the machine learning (blue) and linear (orange) estimates
derived from the S-index and TSI. The time series for the three filling factors are plotted in the left column. The estimated filling factors are
plotted as a function of the HMI filling factors in the right column—the grey dashed lines indicate a slope of 1, and are meant to guide the
eye. Both the linear and machine learning techniques reproduce the directly-observed values of fspot ,fplage, and fntwk . Note that there is a slight
offset between the linear estimate of fnt wk and the SDO measurements. However, this offset is well within the expected 20% - 50% definitional
variations reported by Meunier et al. (2010)
While plage and network occupy a greater area than spots
on the Sun, and therefore dominate the suppression of con-
vective blueshift, the higher brightness contrast of spots
means that they drive the photometric RV shift, ∆vphot . We
expect ∆vphot to scale with number and size of the spots ro-
tating across the solar surface. However, we also expect a
phase lag between fspot and ∆vphot . For a single spot moving
across the solar disk, fs pot is at its maximum value when the
spot is on the center of the solar disk. However, the absolute
value of ∆vphot is maximized when the spot is at the solar
limb, rotating toward or away from the observer, and is zero
when the spot is at disk center. We therefore expect ∆vphot to
also depend on the derivative of the filling factor with respect
to time:
∆vphot ∝fspot ×d fs pot
dt .(14)
Note that this formulation mirrors the FF0method developed
by Aigrain et al. (2012).
By combining Eqs. 12,13, and 14, we therefore produce
a model of the solar RVs based on our estimated feature-
specific magnetic filling factors:
RV =A fspot d fspot
dt +B fs pot +C fpl age +D fnetwork +RV0.
(15)
Note that the offset Ein Eq. 13 has been absorbed into RV0.
10 T. MILBO URNE ET AL.
Figure 5. 230 day subsets of the time series of the SDO/HMI-observed magnetic filling factors (black), along with the MLP (blue) and linear
(orange) estimates derived from the S-index and TSI. Three subsets are shown, taken during the middle of the stellar cycle (left), during solar
maximum (middle), and approaching solar minimum (right). Both techniques successfully reproduce fspot ,fpl age, and fnt wk, with especially
good performance at solar minimum.
We then fit the HARPS-N solar telescope RVs to Eq. 15
using the directly-measured SDO filling factors. The results
of each fit is given in Table 7. Note that since the linear
estimates of fbright,fnt wk, and fplage are linear transformations
of the S index, as shown in Eqs. 6,9, and 10, we require D= 0
to avoid degeneracies when using the filling factor estimates
derived from the linear technique. Fitting the linear estimated
filling factors to Eq. 15 without this constraint is equivalent
to fitting to RV =A fspot d fspot
dt +B fs pot +C0SHK +D0SHK +E0.
We therefore set D= 0 to avoid having degeneracies between
Cand Din our fit. No such constraint on Dis necessary when
considering the SDO or MLP filling factors.
To ensure our fit is indeed reproducing the suppression of
convective blueshift and the photometric RV shift, as ex-
pected, we compare the relevant terms of Eq. 15 to the
SDO/HMI estimates of these RV perturbations, as calculated
ESTIMATING MAGNET IC FILLING FAC TORS FROM SIMULTANEOUS SPECTROSCOPY AND PHOTOMETRY 11
RMS (m s−1)
Full solar dataset 1.64
Decorrelated with S index 1.10
Decorrelated with HMI filling factors 0.91
Decorrelated with linear filling factor estimates 1.04
Decorrelated with MLP filling factor estimates 1.02
Decorrelated with MLP RV estimate 0.96
Table 6. RMS RV residuals from several models and methods. Us-
ing our estimates of fs pot,fpl age, and fntwk in Eq. 15 reduces the RMS
RVs by 60 cm s−1. However, using the HMI-observed filling factors
reduces the RMS residuals by a further 13 cm s−1, indicating there
is additional information in these filling factors not captured by our
estimates. A direct MLP fit to the solar RVs, using the S index and
TSI as inputs, performs better than our estimated filling factors, but
does not perform as well as the fit to HMI filling factors. This in-
dicates that, while our estimated filling factors are highly correlated
with the observed values, the S index and TSI alone are insufficient
to completely characterize the filling factors of each feature.
in MH19. In Tables 4and 5, we compare the estimates of
∆vconv and ∆vphot computed from Eqs. 13 and 14 using
the filling factors measured by SDO, and estimated using
the linear and MLP techniques to the values of ∆vconv and
∆vphot derived from HMI observations in Haywood et al.
(2016) and MH19. We see that all of the estimated values
of ∆vconv are highly correlated with the HMI-derived veloc-
ities. Our estimates of ∆vphot are less correlated with the
actual photometric shift, but still show good agreement. In-
terestingly, including the contributions of plage and network
regions in Eq. 14 — that is, adding terms ∝fplage ×d fpl age
dt
and ∝fntwk ×d fntwk
dt — does not appear to increase the cor-
relation coefficient. However, we may still conclude that
the RVs calculated using Eq. 15 indeed do correspond to the
combination of suppression of convective blueshift and pho-
tometric RV shift described by Eq. 12.
For both the SDO measured and MLP-derived filling fac-
tors, we see C>D. This is consistent with the idea that the
denser magnetic interconnections available in photospheric
plages are more successful in inhibiting convection, and thus
convective blueshifts, than the sparser network magnetiza-
tions, as suggested in MH19. Indeed, we see that, using
MLP estimates, the network contribution is consistent with
zero, and using SDO observations, the network contribution
is only ∼2σabove zero.
The Bcoefficients, which describe the spot contributions
to the suppression of convective blueshift, vary depending
on the filling factors used. The linear and MLP estimates of
fspot receive a heavy weighting, while the SDO weighting is
about a factor of 3 smaller. The MLP estimates also have a
contribution only ∼2σabove zero. However, the Sun is a
plage-dominated star, and fspot is about a factor of 100 times
smaller than fplage , as shown in Fig. 2. So, while the precise
weighting of fspot varies based on the values used, in all cases
their contribution to the suppression of convective blueshift
will be negligible compared to that of fplage . We may there-
fore conclude that, as suggested by MH19, plage regions are
the dominate contribution to the solar suppression of con-
vective blueshift, while spots are the dominant contribution
to the photometric RV shift: knowledge of the plage and spot
filling factors are therefore sufficient to reproduce ∆vconv and
∆vphot respectively.
Fitting to the HMI-observed filling factors reduces the
HARPS-N solar RV residuals from 1.64 m s−1to 0.91 m s−1,
as shown in Table 6. In comparison, the usual technique
of simply decorrelating the S-index from the RV measure-
ments (i.e., fitting RV =ASHK +B) results in an RMS of only
1.10 m s−1, indicating that spots, plage, and network regions
have different contributions to the S index, and have different
effects on the suppression of convective blueshift (Meunier
et al. 2010, Fig.7).
Repeating this fit with both our linear and MLP estimates
of fspot ,fntwk , and fplage reduces the RMS RV to 1.04 m s−1
and 1.02 m s−1respectively. This implies that, while our esti-
mates are highly correlated with the true values of the filling
factors, there is additional information in the true filling fac-
tors that is not captured by either technique, resulting in less
precise estimates of the convective blueshift and photometric
RV shifts. Interestingly, while the linear filling factor esti-
mates cannot distinguish the RV contributions of the spots
and network, as discussed above, the linear and MLP esti-
mates result in similar RMS RVs. The fact that the linear es-
timates reduce the RMS RVs below the level obtained from
the S index despite this limitation highlights the importance
of spots in our models of activity-driven RVs.
To see if it is possible for a more refined technique to ex-
tract further RV information from our inputs, we fit an MLP
directly to the solar RVs using the S-index and TSI as inputs.
This is similar to the technique proposed by de Beurs et al.
(2020), but replacing the residual cross-correlation function
with the S-index and TSI. The hyperparameters of this MLP
are the same as those given in Table 2. As before, we divide
our data into training and test sets, and the quoted residuals
are derived from the test set. This fit results in an RMS resid-
ual of 0.96 m s−1, (as shown in Table 6) indicating that there
is indeed more RV information to be gained from this set of
observations. Interestingly, however, while this RMS value
is below the residuals obtained from both sets of estimated
filling factors, it is greater than the 0.91 m s−1residuals ob-
tained by using the direct SDO measurements of the filling
factors. This appears to indicate that, while the S index and
TSI contain more information than our linear and MLP esti-
mates could obtain, they do not contain all the information
about the solar plage, spot, and network coverage.
12 T. MILBO URNE ET AL.
Filling Factor Source A(105m) B(m s−1)C(m s−1)D(m s−1)RV0(m s−1)
SDO 7.7±1.2 165±76 244±15 52 ±27 −2.0±0.3
Linear 7.1±1.3 470±80 281±11 0∗−1.32 ±0.08
MLP 7.5±1.4 479 ±218 242 ±52 2 ±90 −1.3±1.2
Table 7. Best fit coefficients for fitting Eq. 15 to the HARPS-N solar RVs using SDO filling factors, linear estimates of the filling factors, and
MLP estimates of the filling factors. Note that the estimates of fplage and fntwk derived from the linear technique are both linear transformations
of the S index. We therefore require D= 0 to avoid degeneracies when using the filling factor estimates derived from the linear technique.
This is unsurprising: we note that network regions can
form from decaying plage regions. Due to the geometry of
the magnetic flux tubes associated with these regions, a net-
work region may rotate onto the limb, become a plage re-
gion as it rotates onto disk center, and then become a net-
work region again as it rotates back onto the limb. The linear
technique directly uses the different temperature contrasts of
network and plage to provide a useful first pass at differenti-
ating these regions, but does not capture these links between
them. That is, there are additional physical effects that fur-
ther complicate the relationship between photometry, spec-
troscopy, filling factors, and RVs (Miklos et al. 2020). While
the underlying behavior of the MLP is unknown, it likely em-
ploys a similar, slightly more complex technique to differen-
tiate the two classes of regions. The magnetic intensification
effect, which strengthens lines in the presence of a magnetic
field (e.g., Leroy 1962;Stift & Leone 2003), has an RV signal
which depends on the overall filling factor, as well as a given
line’s wavelength, effective Landé gvalue, and the magnetic
field strength (Reiners et al. 2013). HMI monitors the photo-
spheric 6173.3 Å iron line: these wavelength-dependent ef-
fects mean that the filling factors derived from HMI may not
be consistent with those derived from the chromospheric cal-
cium H and K lines. The center-to-limb dependence of the
calcium H and K lines are different than the 6173.3 Å line as
well, which could lead to mismatches in the derived filling
factors as a function of rotational phase. More complicated
linear and MLP-based filling factor estimates could use spec-
troscopic measurements of additional absorption lines, and
photometric measurements integrated over different wave-
length bands to compensate for these effects, and to exploit
different wavelength-dependent contrasts of each feature to
better separate these three classes of magentic active regions.
The direct MLP fit to the solar RVs and its residuals are
plotted in Fig. 6. The effects of HARPS-N cryostat cold plate
warm-ups, discussed in Collier Cameron et al. (2019); Du-
musque et al. (2020), are clearly visible in the fit residuals,
indicating that the MLP is not "learning" instrumental sys-
tematics, and that the residual RV variations below this level
are likely dominated by a combination of instrumental sys-
tematics and activity processes not reflected by variations in
the S-index and TSI. Further work is necessary to identify
Figure 6. MLP fit to the HARPS-N solar telescope data. HARPS-
N RVs are shown in black, and MLP estimates of the RVs are in
blue. Fit residuals are shown in the bottom panels: HARPS-N cryo-
stat warm-up dates (see text) are indicated with black dashed lines.
these remaining activity processes, and to disentangle them
from instrumental effects.
5.2. Application to the Stellar Case
The techniques developed in this work should be applica-
ble to Sun-like stars with the proper observational cadence.
To reproduce properly scaled filling factors, the linear tech-
nique requires precise knowledge of radius, effective tem-
perature, and distance to the target, as well as the tempera-
ture contrasts of the plage, spots, and network. The effective
temperature may be calculated spectroscopically (Buchhave
et al. 2012), while the temperature contrasts may be assumed
to be Sun-like in the case of G-class stars. The stellar ra-
dius may then be calculated photometrically, using the spec-
troscopic temperature as a prior. The stellar distance may
be straightforwardly determined through parallax measure-
ESTIMATING MAGNET IC FILLING FAC TORS FROM SIMULTANEOUS SPECTROSCOPY AND PHOTOMETRY 13
ments, while the rotation period may be obtained via pho-
tometry or through RV measurements.
Although the techniques presented here assume a plage-
dominated star, it is straightforward to rework Eqs. 3and 5
for a spot-dominated target: in this case, the S index is as-
sumed to be correlated with spot-driven variations of the TSI,
with positive deviations indicating the presence of bright,
plage regions.
While properly scaling the linear estimate of fplage requires
observations near stellar minimum, modelling RV variations
only requires time series which are proportional to fspot ,
fplage and fntwk —in this case, this offset is unimportant, and
no additional constraint is placed on the stellar observations.
Furthermore, the values of ∆Tspot ,∆Tplage and ∆Tntwk may
be absorbed into the fit coefficients in Eqs. 3,5, and 15, fur-
ther simplifying matters.
The MLP machine learning technique, in contrast, requires
less knowledge of the target star: while the mathematical and
physical transformations learned by MLP are unknown to
the user, the MLP is presumably learning a more sophisti-
cated version of the linear technique, and implicitly "learns"
the solar values for feature temperature contrast and quiet
temperature as it identifies higher-order correlations between
the TSI, S index, and filling factors. This makes the MLP
straightforward to implement when precise contrast values
are unknown. Furthermore, since the MLP uses no timing
information, it places no constraints on the observational ca-
dence or baseline: it only requires simultaneous photometric
and spectroscopic measurements.
However, since ground-truth filling factors can only be di-
rectly measured in the solar case, the MLP must be trained
using solar data. Its stellar application is therefore limited to
Sun-like stars (that is, stars with very similar surface filling
factors as the Sun, or possibly even only solar twins), mak-
ing it less generalizable to other stellar targets. Since stars
other than the Sun cannot be resolved spatially at high reso-
lution, assessing just how "Sun-like" a target needs to be for
the machine learning technique to yield meaningful results
is challenging. One possibility is to generate synthetic stel-
lar images for targets with a variety of spectral types, activ-
ity levels, feature contrasts, and viewing angles using SOAP
2.0 (Dumusque et al. 2014), StarSim (Herrero, Enrique et al.
2016), or a similar platform, computing the light curve and
S-index for these images, and seeing if an MLP trained on
the solar case reproduces the filling factors expected for each
image. Such an analysis is beyond the scope of this work.
6. CONCLUSIONS
We assess two techniques to extract spot, plage, and net-
work filling factors using simultaneous spectroscopy and
photometry. The first technique involves a straightforward
analytical manipulation of the S-index and TSI time series,
while the second uses a neural network machine learning
technique known as a Multilayer Perceptron (MLP) trained
on ground-truth filling factors derived from full-disk solar
images. Both techniques yield filling factor estimates which
are highly correlated with values derived from full-disk solar
images, with Spearman correlation coefficients ranging from
0.81 and 0.89 from each technique.
We show that decorrelating a nearly-three-year time se-
ries of solar RVs using HMI-observed spot, plage, and net-
work filling factors effectively reproduces the expected RV
variations due to the convective blueshift and rotational im-
balance due to flux inhomogeneities, reducing the residual
activity-driven RVs more than the typical technique of decor-
relating using spectroscopic activity indices alone. Fitting to
HMI filling factors reduces the RV RMS from 1.64 m s−1
to 0.91 m s−1, while fitting to the S-index alone results in
an RMS variation of 1.10 m s−1. Including this additional
information about spots, plage, and network thus accounts
for an additional p(1.10m s−1)2−(0.91m s−1)2= 0.62m s−1
of RMS variation. The filling factor estimates from both the
linear and MLP techniques offer some improvement to the
RMS residuals beyond what is obtained from only the S-
index. Decorrelating with the linear estimates reduces the
RMS variation to 1.04 m s−1, and the MLP estimated filling
factors reduces the RMS to 1.02 m s−1.
Using a MLP trained directly on the solar RVs, we re-
duced the RMS to 0.96 m s−1. While this indicates that the
S-index and TSI contain more RV information than obtained
by either estimate of our filling factors, it does not lower the
RMS RVs below the 0.91 m s−1limit obtained using direct
measurements of the magnetic filling factors. This suggests
that, while our initial estimates of fspot ,fpl age, and fntwk are
highly correlated with the expected value, more information
is needed to fully characterize these feature-specific filling
factors. To match the performance of the HMI filling fac-
tors, a more sophisticated version of this technique, using
additional spectral lines and photometric bands will likely be
necessary.
Both the analytical and machine learning techniques may
be used to extract filling factors on other stars: the analytical
technique is more widely generalizable, but requires detailed
knowledge of the star and good temporal sampling, ideally
with observations of the target at activity minimum. The ma-
chine learning technique, in contrast, requires no additional
knowledge of the target star, and applies no constraints on the
observing schedule—however, it is only applicable to stars
with very similar filling factor properties as the Sun.
14 T. MILBO URNE ET AL.
ACKNOWLEDGMENTS
This work was primarily supported by NASA award num-
ber NNX16AD42G and the Smithsonian Institution. The so-
lar telescope used in these observations was built and main-
tained with support from the Smithsonian Astrophysical Ob-
servatory, the Harvard Origins of Life Initiative, and the
TNG.
This material is also based upon work supported by NASA
under grants No. NNX15AC90G and NNX17AB59G issued
through the Exoplanets Research Program. The research
leading to these results has received funding from the Eu-
ropean Union Seventh Framework Programme (FP7/2007-
2013) under grant Agreement No. 313014 (ETAEARTH).
The HARPS-N project has been funded by the Prodex Pro-
gram of the Swiss Space Office (SSO), the Harvard Univer-
sity Origins of Life Initiative (HUOLI), the Scottish Univer-
sities Physics Alliance (SUPA), the University of Geneva, the
Smithsonian Astrophysical Observatory (SAO), and the Ital-
ian National Astrophysical Institute (INAF), the University
of St Andrews, Queen’s University Belfast, and the Univer-
sity of Edinburgh.
We would like to acknowledge the excellent discussions
and scientific input by members of the International Team,
"Towards Earth-like Alien Worlds: Know thy star, know thy
planet", supported by the International Space Science Insti-
tute (ISSI, Bern).
This work was partially performed under contract with the
California Institute of Technology (Caltech)/Jet Propulsion
Laboratory (JPL) funded by NASA through the Sagan Fel-
lowship Program executed by the NASA Exoplanet Science
Institute (R.D.H.).
SHS is grateful for support from NASA Heliophysics LWS
grant NNX16AB79G.
A.M. acknowledges support from the senior Kavli Institute
Fellowships.
ACC acknowledges support from the Science and Tech-
nology Facilities Council (STFC) consolidated grant number
ST/R000824/1.
X.D. is grateful to the Branco-Weiss Fellowship for contin-
uous support. This project has received funding from the Eu-
ropean Research Council (ERC) under the European Unions’
Horizon 2020 research and innovation program (grant agree-
ment No. 851555).
CAW acknowledges support from Science and Technology
Facilities Council grant ST/P000312/1.
HMC acknowledges financial support from the National
Centre for Competence in Research (NCCR) PlanetS, sup-
ported by the Swiss National Science Foundation (SNSF), as
well as a UK Research and Innovation Future Leaders Fel-
lowship.
We thank the entire TNG staff for their continued support
of the solar telescope project at HARPS-N.
Facilities: TNG: HARPS-N, Solar Telescope
Software: HARPS-N DRS, Python: NumPy,SciPy,
scikit-learn
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