# They Do Change After All: 25 Years of GONG Data Reveal Variation of p-Mode
Energy Supply Rates

René Kiefer, Anne-Marie Broomhall,
Centre for Fusion, Space, and Astrophysics, Department of Physics, University
of Warwick, Coventry, CV4 7AL, United Kingdom E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

It has been shown over and over again that the parameters of solar p modes
vary through the solar activity cycle: frequencies, amplitudes, lifetimes,
energies. However, so far, the rates at which energy is supplied to the p
modes have not been detected to be sensitive to the level of magnetic
activity. We set out to re-inspect their temporal behaviour over the course of
the last two Schwabe cycles. For this, we use Global Oscillation Network Group
(GONG) p-mode parameter tables. We analyse the energy supply rates for modes
of harmonic degrees $l=0\text{--}150$ and average over the azimuthal orders
and, subsequently, over modes in different parameter ranges. This averaging
greatly helps in reducing the noise in the data. We find that energy supply
rates are anti-correlated with the level of solar activity, for which we use
the $F_{10.7}$ index as a proxy. Modes of different mode frequency and
harmonic degrees show varying strengths of anti-correlation with the
$F_{10.7}$ index, reaching as low as $r=-0.82$ for low frequency modes with
$l=101\text{--}150$. In this first dedicated study of solar p-mode energy
supply rates in GONG data, we find that they do indeed vary through the solar
cycle. Earlier investigations with data from other instruments were hindered
by being limited to low harmonic degrees or by the data sets being too short.
We provide tables of time-averaged energy supply rates for individual modes as
well as for averages over disjunct frequency bins.

###### keywords:

Sun: activity – Sun: helioseismology – Sun: oscillations – methods: data
analysis

††pubyear: 2020††pagerange: They Do Change After All: 25 Years of GONG Data
Reveal Variation of p-Mode Energy Supply Rates–LABEL:lastpage

##  1 Introduction

Solar acoustic oscillations (p modes) are stochastically excited in the
turbulent, outer convective layers of the Sun (e.g., Balmforth, 1992; Rimmele
et al., 1995; Houdek & Dupret, 2015). They are well described by a damped and
stochastically forced harmonic oscillator in the time domain (e.g., Anderson
et al., 1990). In turn, in the Fourier domain a single p-mode peak is well
characterised by a Lorentzian profile with a frequency $\nu$, width $\Gamma$,
and height $A$ (neglecting the observed mode asymmetry, e.g., Nigam et al.,
1998; Korzennik, 2017; Philidet et al., 2020). Measurements of these mode
parameters in the Fourier domain can be used to learn about the energetics of
the oscillator in the time domain: the peak width $\Gamma$ is inversely
proportional to the mode’s lifetime and thus holds information about the
damping. The product of peak height – or amplitude – $A$ and $\Gamma$, after
proper scaling with the corresponding mode mass, encodes the energy $E$ that
is stored in the respective p-mode oscillation (e.g., Goldreich et al., 1994;
Baudin et al., 2005). Ultimately, the energy that is supplied to a mode per
unit time $dE/dt$ is proportional to the product $\Gamma^{2}\cdot A$ (e.g.,
Goldreich et al., 1994). This quantity is thus directly connected to the
forcing of the mode (e.g., Chaplin et al., 2003). Possible temporal changes in
the forcing function of the oscillator can thus be measured via time-resolved
measurement of the quantity $\Gamma^{2}\cdot A$.

Magnetic activity affects p modes in various ways: frequencies of modes below
the acoustic cut-off (Balmforth & Gough, 1990), i.e., modes that are
essentially trapped in the solar interior, are correlated with the level of
magnetic activity: frequencies are highest during times of strong solar
activity (e.g., Woodard & Noyes, 1985; Elsworth et al., 1990; Libbrecht &
Woodard, 1990; Jiménez-Reyes et al., 1998; Chaplin et al., 2001; Salabert et
al., 2015; Tripathy et al., 2015; Broomhall, 2017). For oscillations above the
acoustic cut-off – these are waves that escape into the solar atmosphere
(Balmforth & Gough, 1990; Fossat et al., 1992; Vorontsov et al., 1998) –
frequencies turn anti-correlated with the activity cycle (see, e.g., Woodard &
Libbrecht, 1991; Howe et al., 2008; Rhodes et al., 2010). The magnitude of
activity-related shifts of mode frequencies depend on both mode frequency and
harmonic degree $l$ (see Basu 2016 and references therein).

On the energetic side, p modes are increasingly damped with more activity on
the surface, which leads to broader peak widths, as the two are inversely
related to each other (e.g., Jefferies et al., 1991; Komm et al., 2000b;
Chaplin et al., 2000; Jiménez et al., 2002; Jiménez-Reyes et al., 2003;
Jiménez-Reyes et al., 2004; Salabert et al., 2007; Burtseva et al., 2009;
Broomhall et al., 2015; Kiefer et al., 2018 and see Broomhall et al. 2014 for
a more extensive list of references on the matter). The observed changes along
the solar cycle are measured to be between 10–20 depending on mode frequency
and harmonic degree (e.g., Kiefer et al., 2018). Mode amplitudes are anti-
correlated with activity (e.g., Pallé et al., 1990; Elsworth et al., 1993;
Chaplin et al., 2000; Komm et al., 2000b; Jiménez et al., 2002; Jiménez-Reyes
et al., 2003; Jiménez-Reyes et al., 2004; Broomhall et al., 2015; Kiefer et
al., 2018). Again, the magnitude of these changes is somewhat dependent on
mode frequency and harmonic degree (e.g., Kiefer et al., 2018) but are
typically between 10–25 over the solar cycle. As mode amplitudes can be
converted into mode energies, these are also anti-correlated with the level of
activity.

The energy supply rate (or excitation rate, forcing rate, energy dissipation
rate through the oscillator) of the p modes has thus far been assumed to be
constant: This has been explained on the simple reason that changes in damping
are of the same magnitude as the accompanying changes in mode energies. Thus,
changes in mode damping are sufficient to explain the entire observed
phenomenology (Chaplin et al., 2000): modes get broader and their amplitudes
get smaller. In turn, the increased damping explains the lower mode energies
during phases of higher activity without having to propose a variation to the
forcing function. For more details on how mode damping, excitation, and
forcing are related under the assumption of a harmonic oscillator, we refer to
Chaplin et al. (2000), in particular their Section 2.

In addition, measurements of the energy supply rates – via the compound
quantity $\Gamma^{2}\cdot A$ – have repeatedly returned constancy through the
solar activity cycle (Chaplin et al., 2000; Jiménez-Reyes et al., 2004;
Jiménez-Reyes et al., 2003; Salabert et al., 2007; Broomhall et al., 2015).
However, all of these studies investigated only low harmonic degrees $l\leq 3$
from disk-integrated Sun-as-a-star data.

The energy supply rate of p modes is intimately connected with the spatial and
temporal properties of convection (e.g., Samadi et al., 2003). The temporal
correlation of convective eddies is crucial to the theoretical understanding
and modelling of mode energy supply rates (e.g., Houdek, 2010). Different
analytical descriptions of the eddy-time-correlation can lead to vastly
different values in the predicted energy supply rates (e.g., Belkacem et al.,
2010). Given the observed changes of near-surface convection properties
through the solar cycle, e.g., a decrease in the size of granules (Macris et
al., 1984; Muller, 1988) and decreasing granular contrast with increasing
level of magnetic activity (Muller et al., 2007), it can be expected that the
energy supply rates are in fact not entirely constant through the solar
activity cycle.

Short term departures from constancy have already been detected for disk-
integrated helioseismic observations before: Chaplin et al. (2003) report on a
roughly $100\text{\,}\mathrm{d}$ long augmentation of the mode forcing during
1998 in Birmingham Solar Oscillation Network (BiSON) data.

A hint in GONG data towards activity-related changes of p-mode energy supply
rates was found by Komm et al. (2000a) but has never been substantiated. Also,
Komm et al. (2000c) noted that energy supply rates decrease with increasing
activity, but the detected change amounted to a mere $-4.4\pm
4.3\text{\,}\mathrm{\char 37\relax}$ for the average over modes with $15\leq
l\leq 95$ and
$$1.6\text{\,}\mathrm{mHz}$\leq\nu\leq$3.1\text{\,}\mathrm{mHz}$$. Thus, no
long-term variation of the mode energy supply rates that is (anti-)correlated
with the level of magnetic activity has been conclusively confirmed as yet.

In the following we describe the data we used in this study (Section 2) and
how we prepared and corrected these data (Section 3). The results from our
analyses are presented in Section 4. A detailed discussion follows in Section
5, closing with the conclusions we draw from this study and its findings in
Section 6.

Figure 1: $\overline{F_{10.7}}$ solar radio flux during the GONG operation. The blue horizontal line indicates the median value. Times of activity maxima and minima are highlighted by the grey areas. Individual data points are averages of daily $F_{10.7}$ values over the 108 days of the GONG datasets. Table 1: Start and end dates of solar activity extrema, mean $\overline{F_{10.7}}$ index during, and number of GONG datasets during these times. Extremum | Start | End | $\overline{F_{10.7}}$ | #GONG
---|---|---|---|---
|  |  | $\left[$\mathrm{s}\mathrm{f}\mathrm{u}$\right]$ | datasets
Minimum 23 | 1995-05-07 | 1997-08-11 | 73.7 | 23
Maximum 23 | 2000-01-29 | 2002-05-05 | 184.9 | 23
Minimum 24 | 2007-09-21 | 2009-11-25 | 70.0 | 22
Maximum 24 | 2012-01-01 | 2014-04-30 | 126.8 | 24
Minimum 25 | 2018-01-01 | 2020-04-12 | 70.4 | 19

##  2 Data

###  2.1 GONG mode parameters

At the time of writing, the publicly available GONG mode parameter files cover
more than one full cycle of the solar magnetic field, i.e., more than two
11-year Schwabe cycles: the start date of the time series is May 7, 1995, the
last available segment ends on April 12, 2020. We use the p-mode parameter
tables, which were produced by the standard GONG pipeline from solar full-disk
Dopplergrams (Anderson et al., 1990; Hill et al., 1996; Hill & Howe, 1998) and
are available online111ftp://gong2.nso.edu/TSERIES/v1f. The parameters in
these tables are obtained by fitting the power spectra of 108-day long GONG
Doppler-velocity time series. Consecutive datasets overlap by 72 days, i.e.,
each third dataset is independent. Every 36 days a new dataset is added, which
defines the unit of one GONG month. Thus, over one year about ten GONG months
are accumulated.

In each GONG dataset’s power spectrum, the oscillation multiplets are fitted
with one symmetrical Lorentzian profile per azimuthal order $-l\leq m\leq l$
and a linear background for all harmonic degrees up to $l=150$. The free
parameters of the fit model are mode frequency $\nu_{nlm}$, mode width
$\Gamma_{nlm}$ (full width at half maximum), mode amplitude $A_{nlm}$,
background offset $b_{0,nlm}$, and background slope $b_{1,nlm}$, where the
triple $\left(nlm\right)$ indicates the radial order $n$, harmonic degree $l$,
and azimuthal order $m$ of a mode. In this article, we concentrate on the
product $\Gamma_{nlm}^{2}\cdot A_{nlm}$, which is, as we will see in further
detail in Section 3.2, proportional to the mode energy supply rate
$dE/dt_{nlm}$.

###  2.2 Solar radio flux $F_{10.7}$

As a proxy for solar magnetic activity, we used the solar radio flux given by
the $F_{10.7}$ index (Tapping, 2013), which is a good proxy for the level of
activity in the upper chromosphere and the lower corona (Tapping, 1987;
Broomhall & Nakariakov, 2015). Its time series is available
online222ftp://ftp.seismo.nrcan.gc.ca/spaceweather/solar_flux/daily_flux_values/.
For a measurement of $F_{10.7}$, the total emission on the solar disk at a
wavelength of $10.7\text{\,}\mathrm{cm}$ is integrated over one hour. It is
given in solar flux units $\mathrm{s}\mathrm{f}\mathrm{u}$, where
$1\text{\,}\mathrm{s}\mathrm{f}\mathrm{u}$ =
${10}^{-22}\text{\,}\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{H}\mathrm{z}^{-1}$.
The $F_{10.7}$ index is the averaged $F_{10.7}$ flux scaled for a distance of
$1\text{\,}\mathrm{A}\mathrm{U}$.

There is no continuous time series that covers the entire time period that
GONG has been in operation. Thus, we concatenate the tables
F107_1947_1996.txt, F107_1996_2007.txt, and fluxtable.txt (data since 2004),
which are available at the given URL, at the end date of each preceding table,
ignoring the overlapping data. Measurements which are taken on the same day
are averaged. These daily values were then averaged over the 108-day periods
of the GONG datasets to yield $\overline{F_{10.7}}$.

In Figure 1, the resulting $\overline{F_{10.7}}$ index is shown as a function
of time, limited to the period covered by the available GONG data. The blue
horizontal line indicates the median value of the $\overline{F_{10.7}}$ index
through the used time frame, $\overline{F_{10.7}}_{\text{,
median}}=$95.6\text{\,}\mathrm{s}\mathrm{f}\mathrm{u}$$. Data points above
this value are coloured red, points below are black. We will use this colour
code in later figures to ease the interpretation of the data.

The start and end dates of the periods of solar activity minima and maxima
through the GONG observations are listed in Table 1. These periods are shaded
grey in Figure 1 and the respective identifier is printed at the top of the
panel. Again, we will reuse this background colouring in later figures to ease
identification of activity minima and maxima. The dates for the first three
extrema are taken from Broomhall (2017). The dates of Max24 were chosen such
as to have a similar number of GONG datasets as the preceding maximum at times
of high activity. The time when $\overline{F_{10.7}}$ has reached a comparable
level of low activity as Min24 was chosen as the starting date of the
currently ongoing minimum. The number of GONG datasets during each of these
periods are given in the last column of Table 1.

##  3 Data Preparation

The data processing steps and the corrections that were applied to the
quantity $\Gamma_{nlm}^{2}\cdot A_{nlm}$ are the same as described in detail
by Komm et al. (2000b, c), and Kiefer et al. (2018) for its constituent
quantities $\Gamma_{nlm}$ and $A_{nlm}$. Here, we will briefly lay out the
rationale of these corrections and refer to Kiefer et al. (2018) for more
details.

###  3.1 Corrections and averaging

#### Correction for spatial masking and azimuthal averaging

Due to projection effects that increase towards the solar limb – affecting
pixel resolution as well as measured radial velocities – mode amplitudes of
azimuthal orders with lower $|m/l|$ are suppressed. This suppression follows
an empirically determined polynomial in $\left(m/l\right)^{k}$ with $k=0,2,4$.
To correct for this, we subtracted the fit from each multiplet and added the
value of $|m/l|=1$.

This fit, as all linear regressions in this article, was performed with
statsmodels’s weighted linear regression routine (Seabold & Perktold, 2010).
In this particular case, the fit was done multilinearly in
$\left(m/l\right)^{2}$ and $\left(m/l\right)^{4}$. The variance-weighted mean
of the corrected data points is then adopted as the representative value of
$\Gamma_{nl}^{2}\cdot A_{nl}$ of each multiplet. For a multiplet to be
considered at all, a minimum of one third of azimuthal orders was required to
be present for multiplets with $l>10$ and two thirds for multiplets with
$l\leq 10$.

In the following, the index $m$ is dropped from all quantities, as the
azimuthal orders of a multiplet have been averaged over. The frequency of the
$m=0$ singlet is taken as the multiplet frequency. If the $m=0$ was not
fitted, then the mean of the $m=\pm 1$ singlet frequencies is used. As the
azimuthal orders have been averaged over, from here on out the phrase
multiplet will be avoided where possible and the term mode will be used
instead.

#### Correction for temporal window

GONG is a network of six observing stations that are distributed around the
Earth to maximise continuous viewing of the Sun. Still, the duty cycle
(henceforth called fill) of the time series is lower than unity for all GONG
datasets.333https://gong2.nso.edu/fill.txt The fill of the 108-day time series
varies between a minimum of $69.7\text{\,}\mathrm{\char 37\relax}$ and a
maximum of $93.7\text{\,}\mathrm{\char 37\relax}$ with a median value of
$86.7\text{\,}\mathrm{\char 37\relax}$. This temporal window function
artificially increases mode widths and decreases mode amplitudes.

As the fill is rather high and the gaps are typically not ordered in a
repeating pattern, it is sufficient to account for its impact by a linear
regression (Komm et al., 2000b): $\Gamma_{nl}^{2}\cdot A_{nl}$ of each mode
are fitted with a linear function as a function of fill, which is then
subtracted and the extrapolated value at fill $=1$ is added.

#### Apparent solar radius

Just as Kiefer et al. (2018), we corrected for residual annual changes of the
solar angular radius. For this, we performed a linear fit of
$\Gamma_{nl}^{2}\cdot A_{nl}$ as a function of the solar apparent radius and
adjusted $\Gamma_{nl}^{2}\cdot A_{nl}$ to its extrapolated value at the
maximum of the apparent solar radius during the time series.

#### Jumps in mode amplitudes

Kiefer et al. (2018) discovered two jumps in the mode amplitudes, which had to
be corrected for with an empirical correction factor. The first jump occurs
around the year 2001 and is due to an upgrade of the GONG hardware. The second
jump, around 4 years later, is of uncertain origin. We continue using the
correction factor of Kiefer et al. (2018), which they give in their appendix
A, equation (12).

Figure 2: Variation of energy supply rate parameter $\Theta$ as a function of
time and mode frequency (left panel) and of harmonic degree (right panel).
$\Theta_{nl}$ of each mode is first normalised to the mean over the entire
time series, then averages over independent bins of
$50\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ (in the left panel) or individual
harmonic degree (right panel) are calculated. The colour map is capped at
$\left[0.9,1.1\right]$.

###  3.2 Energy supply rates

Subsequently, we will consider either the energy supply rate parameter of
individual modes, which, for brevity, we baptise
$\Theta_{nl}=\Gamma_{nl}^{2}\cdot A_{nl}$, or the physical quantity of energy
supply rate $dE/dt_{nl}$, see Eq. (1). If the average over modes in certain
ranges of mode frequency and harmonic degrees is taken, we drop the indices
$\left(nl\right)$.

#### Maps of the temporal variation of $\Theta$

Figure 2 shows the normalised variation of $\Theta$ as a function of time and
mode frequency (left panel) and harmonic degree (right panel). Here, the
modes’ $\Theta_{nl}$ are first normalised to the mean over the entire time
series. In the left panel, modes are grouped in independent bins of
$50\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ and averaged. In the right panel,
the average is taken for all modes of the same harmonic degree. The colour map
is capped at $[0.9,1.1]$.

It can be seen that there is considerable temporal variation in $\Theta$, with
bluer, i.e., larger values during activity minima as stated in Table 1. Red
values, which indicate smaller $\Theta$, can predominantly be found at times
of high magnetic activity.

The rather sharp blue vertical feature at around the year 2001 is due to the
GONG hardware upgrade at the time and can be considered an instrument
artefact.

From these two panels, it can already be estimated that any cyclic variation
in the mode energy supply rates is likely less apparent in modes with
frequencies $\gtrsim$3500\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ and
harmonic degrees $\gtrsim~{}120$. Also, even though the variation with
apparent solar radius has been accounted for, a seemingly repeating, quasi-
yearly pattern can be identified. We will discuss this further in Section 5.

Figure 3: $l$-$\nu$-diagram of modes which are present in at least
$50\text{\,}\mathrm{\char 37\relax}$ of all time samples. The colour map gives
the decadic logarithm of the time averaged energy supply rate. Mode inertiae
are normalised at the GONG observation height $r_{\text{obs}}$. The vertical
and horizontal dashed grey lines indicate the boundaries between mode sets
which we use in later sections.

#### Conversion to energy per unit time

From the measured quantity $\Theta_{nl}=\Gamma_{nl}^{2}\cdot A_{nl}$, the
energy that is supplied to the p-mode oscillations per mode can be calculated
by (see, e.g., Goldreich et al., 1994)

$\displaystyle\left(\frac{dE}{dt}\right)_{nl}=\pi^{2}C_{\text{vis}}M_{nl}\Gamma_{nl}^{2}A_{nl},$
(1)

where $\Gamma_{nl}$ is in units of $\mathrm{Hz}$, $A_{nl}$ has units
$\mathrm{c}\mathrm{m}^{2}\mathrm{s}^{-2}\mathrm{Hz}^{-1}$. The numerical
factor $C_{\text{vis}}=3.33$ corrects for the GONG specific reduced visibility
of modes due to leakage effects (Hill & Howe, 1998). We will comment on the
use of this factor in Section 5.3. The factor $\pi^{2}$ stems from the
Lorentzian profile that is used to fit the mode peaks and the assumption of
scattering being the main contribution to mode damping. The mass $M_{nl}$ of
the mode $\left(nl\right)$ is calculated as

$\displaystyle M_{nl}=4\pi M_{\odot}I_{nl},$ (2)

with the solar mass $M_{\odot}$ in $\mathrm{g}$ and the mode inertia of solar
Model S (Christensen-Dalsgaard et al., 1996), which is defined as

$\displaystyle
I_{nl}=\frac{\int_{0}^{R_{\odot}}\rho\left(\left|\xi^{r}_{nl}\right|^{2}+l\left(l+1\right)\left|\xi^{h}_{nl}\right|^{2}\right)r^{2}dr}{M_{\odot}\left(\left|\xi^{r}_{nl}\left(r^{\prime}\right)\right|^{2}+l\left(l+1\right)\left|\xi^{h}_{nl}\left(r^{\prime}\right)\right|^{2}\right)},$
(3)

where $\rho$ is density, $\xi^{r}_{nl}$ and $\xi^{h}_{nl}$ are the mode’s
radial and horizontal displacement eigenfunctions, $R_{\odot}$ is the solar
radius, $r$ is the radial coordinate, and $r^{\prime}$ is the radial position
at which the normalisation is done. We normalise the mode inertiae at either
the photospheric radius $r^{\prime}=R_{\odot}$ or the GONG observation height
at $r^{\prime}=R_{\odot}+$240\text{\,}\mathrm{k}\mathrm{m}$\equiv
r_{\text{obs}}$ (Baudin et al., 2005). Unless stated otherwise, mode inertiae
are normalised at the GONG observation height $r_{\text{obs}}$. Values
normalised at $R_{\odot}$ are provided in an online-supplement table.

We use $\mathrm{c}\mathrm{g}\mathrm{s}$ units, i.e., $dE/dt_{nl}$ has units
$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$. More detailed discussions
of how Eq. (1) comes about can be found in, e.g., Goldreich et al. (1994) and
Komm et al. (2000c). We note that normalised variations of $dE/dt_{nl}$ as a
function of time are equivalent to normalised variations of $\Theta_{nl}$ in
time, as the factors $\pi^{2}C_{\text{vis}}M_{nl}$ then cancel.

Figure 3 shows an $l$-$\nu$-diagram of the 1580 modes which are present in at
least $50\text{\,}\mathrm{\char 37\relax}$ of the GONG datasets. We excluded
modes with $l=0,1$ as they appear as outliers in the distribution of supply
rates (see Figure 4). There are 26 radial and dipole modes, i.e., the total
number of modes with a presence rate of at least $50\text{\,}\mathrm{\char
37\relax}$ is 1606. The colour of the points indicates
$\log_{10}\left(\overline{dE/dt_{nl}}\right)$, where the variance-weighted
average over all time samples (indicated by the overline) is taken as the
representative value for each mode. The vertical and horizontal dashed grey
lines indicate the boundaries in mode frequencies and harmonic degrees that
separate mode sets which we use in later sections.

The top left panel of Figure 4 shows the energy supply rates of all modes as a
function of mode frequency. Again, the variance-weighted average over all time
samples is taken. Squares highlight radial modes, triangles are used to accent
dipole modes. Here it can clearly be seen that radial and dipole modes are
outliers from the general bulk of data. This problem arises from the non-
inclusion of the instrument-specific leakage matrix in the GONG fitting
pipeline. We do not adopt individual correction factors for different harmonic
degrees in this article (consistently with Komm et al. 2000b). From here on
out we shall not consider radial and dipole modes any more.

In the bulk of the data points, ridges can be identified. These are made up of
modes of like radial order $n$, as is highlighted by the discrete colour map.
These ridges can be seen more clearly when $\overline{dE/dt_{nl}}$ is plotted
as a function of harmonic degree $l$, see the bottom left panel of Figure 4.

In Table 5 we give the energy supply rates for averages over 32 modes
consecutive in mode frequency. The second and third columns of this table are
energy supply rates without and with rescaling by $Q$ averaged over the entire
time series. From the second column we find that energy supply rates peak at
around $3460\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ with a value of
$(13.394\pm
0.004)\text{\times}{10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$.
The entries in the following columns are averaged over periods of activity
extrema, i.e., all activity minima or maxima as defined in Table 1, and scaled
by $Q$ in the last two columns. This table is also available in machine-
readable form online.

#### Accounting for different mode inertiae

As was shown by, e.g., Komm et al. (2002), when plotted as a function of mode
frequency the ridges of individual radial orders are collapsed almost
completely into one ridge by multiplication of $\overline{dE/dt_{nl}}$ with
the scaling factor

$\displaystyle Q_{nl}=\frac{I_{nl}}{\overline{I_{n0}}\left(\nu_{nl}\right)}.$
(4)

Here, the radial mode inertia $I_{n0}$ is interpolated to the mode frequency
$\nu_{nl}$ to obtain $\overline{I_{n0}}(\nu_{nl})$.

The right column of panels in Figure 4 shows the time-averaged energy supply
rate values as in the left column, but scaled by multiplication of $Q_{nl}$.
Now, as can be seen in the top right panel of Figure 4, the supply rates can
well be described by a function which depends only on mode frequency $\nu$. We
used a non-linear least squares optimization to fit the exponential

$\displaystyle
G\left(\nu,p\right)=a+b\cdot\exp\left(-\frac{\left(\nu-\nu_{0}\right)^{2}}{2\gamma^{2}}\right)$
(5)

to the base-10 logarithm of the resulting energy supply rates. This fit is
shown by the solid red line. In Eq. (5), the parameter tuple $p$ comprises:
offset $a$, magnitude $b$, central frequency $\nu_{0}$, and width of the
exponential $\gamma$. The best-fit parameters and their standard uncertainties
are given in Table 2. Though present in the panel, the radial and dipole modes
are excluded from the fit.

We also tested an asymmetric and a symmetric Voigt profile, both of which had
a comparable goodness-of-fit as the exponential of Eq. (5). Larger deviations
from the exponential behaviour occur at low mode frequencies
$\lesssim$1800\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ and at high mode
frequencies $\gtrsim$3600\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$. At low
frequencies, this is at least partially caused by the limited frequency
resolution of the 108-day time series which is
$\approx$0.107\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$. Thus, only a few
frequency bins are available per mode width which become smaller with
decreasing mode frequency, see, e.g., Komm et al. (2000c) and Kiefer et al.
(2018). This can lead to an imprecise estimation of mode widths and thus also
of $\overline{dE/dt_{nl}}$. At high frequencies
$\gtrsim$3600\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ some radial orders
depart from the monotonic increase with mode frequency. This can be
appreciated in the bottom panels of Fig. 4, where the data points with
$n\gtrsim 20$ show a dip in $\overline{dE/dt_{nl}}$.

In a machine-readable online-only table, we provide the energy supply rates of
all 1606 modes which are present in at least $50\text{\,}\mathrm{\char
37\relax}$ of the GONG samples. There, we provide mode inertiae and scaling
factors $Q$ normalised at both the photospheric radius $R_{\odot}$ and the
GONG observation height in the solar atmosphere $r_{\text{obs}}$. The energy
supply rates are then provided for 12 different averages: averaged over all
time samples with and without scaling by $Q$, averaged over times of the
activity minima and maxima, both with and without $Q$-scaling, and all of
these at both normalisation heights $R_{\odot}$ and $r_{\text{obs}}$.

Figure 4: Left column: Energy supply rates calculated with Eq. (1) as function of mode frequency (top panel) and harmonic degree (bottom panel). Right column: As left column, but supply rates are inertia-scaled by multiplication with Eq. (4). Top right panel: The solid red line is a fit of the exponential function Eq. (5) to the supply rates. All panels: Radial orders are indicated by the colours of the data points. Radial and dipole modes are highlighted by red-bordered triangles and squares, respectively. Table 2: Fit parameters and uncertainties of exponential function Eq. (5) to the inertia corrected energy supply rates. | $a$ | $b$ | $\nu_{0}$ | $\gamma$
---|---|---|---|---
|  |  | $\left[$\mathrm{\SIUnitSymbolMicro Hz}$\right]$ | $\left[$\mathrm{\SIUnitSymbolMicro Hz}$\right]$
Value | 17.22 | 5.79 | 3446.7 | 1913.1
Uncertainty | 0.14 | 0.14 | 3.1 | 29.3

##  4 Results

###  4.1 Averages over parameter ranges

As can already be appreciated from Figure 2, there is a temporal variation in
the energy supply rates. In order to decrease the uncertainties, we take
variance-weighted averages over four mode frequency ranges and five ranges in
harmonic degree. These ranges are given in the first four columns of Table 3
and are indicated by grey dashed lines in Figure 3.

As we set the presence rate of modes to $50\text{\,}\mathrm{\char 37\relax}$,
not all modes are present for all time samples. The range of number of modes
within each parameter range that is averaged over is given in the fifth column
of Table 3.

To select the modes which contribute to the variance-weighted average within a
parameter range we do the following: First, we select all the modes, which are
within the $\nu_{\text{min}}$-$\nu_{\text{max}}$ and
${l_{\text{min}}\text{-}l_{\text{max}}}$ ranges. The energy supply rates of
each mode are then normalised to the variance-weighted mean over the entire
time series. Modes which do not satisfy the $50\text{\,}\mathrm{\char
37\relax}$ quota are eliminated from the set. At each time sample we then
remove all modes whose normalised variation is not within the interval
$\left[0.5,1.5\right]$, as very large deviations from 1 are not expected to
occur. Further, we remove all modes that deviate by more than four times the
standard deviation of the contributing modes at each time sample. These two
outlier rejections only remove a handful of modes per time sample, if any.

The variance-weighted averages of the normalised variation of $dE/dt$ over
these mode sets are presented in Figure 5. Parameter ranges of the mode set
entering each panel are indicated to the top of each column and the right of
each row. The error bars indicate $5\sigma$, i.e., uncertainties on the
individual data points are very small due to the averaging over azimuthal
orders and the modes in each range.

The minimum and maximum variations of each panel are given in the sixth and
seventh column of Table 3. To reduce short-term variation caused by seasonal
variations in data quality that are not captured by the fill factor
correction, these values are calculated from the one-year boxcar smoothed
data. The median uncertainty of the unsmoothed data is given in the last
column of Table 3. The temporal variation of $dE/dt$ is statistically
significant in all 20 parameter ranges.

Figure 5: Temporal variation of energy supply rates for different ranges of harmonic degrees (rows; harmonic degree range indicated to the right of the fourth column) and mode frequencies (columns; frequency range indicated above the first row). Mode energy supply rates are normalised to the mean for each mode and then the variance-weighted average over all modes in the respective range of frequency and degree is taken. Data with higher than median $\overline{F_{10.7}}$ solar radio flux are highlighted by red points. Levels of 1.1 and 0.9 of the mean are indicated by dashed lines. Grey shaded times corresponds to the activity minima and maxima laid out in Figure 1 and Table 1. Table 3: Normalised and averaged variation of energy supply rates for modes sets with different parameter ranges (defined in columns 1–4) that are presented in the panels of Fig. 5. Column 5 gives the range of number of modes included in the GONG datasets. Columns 6 and 7 give the extrema of the one-year smoothed variation and their uncertainty. The last column gives the median error of the individual un-smoothed energy supply rate in this parameter range. Frequency range [$\mathrm{\SIUnitSymbolMicro Hz}$] | Harmonic degrees | Number of | Normalised amplitude [%] | Median error
---|---|---|---|---
$\nu_{\text{min}}$ | $\nu_{\text{max}}$ | $l_{\text{min}}$ | $l_{\text{max}}$ | modes | min | max | [%]
1365 | 2400 | 2 | 30 | 94–150 | 90.0$\pm$0.1 | 109.3$\pm$0.1 | 0.4
2400 | 3300 | 2 | 30 | 166–185 | 92.9$\pm$0.1 | 109.8$\pm$0.1 | 0.2
3300 | 4360 | 2 | 30 | 125–150 | 94.1$\pm$0.1 | 107.1$\pm$0.1 | 0.3
1365 | 4360 | 2 | 30 | 412–482 | 92.8$\pm$0.1 | 108.4$\pm$0.1 | 0.2
1365 | 2400 | 31 | 60 | 101–137 | 89.2$\pm$0.1 | 110.5$\pm$0.1 | 0.3
2400 | 3300 | 31 | 60 | 94–152 | 89.2$\pm$0.1 | 116.6$\pm$0.1 | 0.2
3300 | 4360 | 31 | 60 | 119–128 | 90.1$\pm$0.1 | 107.4$\pm$0.1 | 0.2
1365 | 4360 | 31 | 60 | 357–417 | 89.8$\pm$0.1 | 111.4$\pm$0.1 | 0.1
1365 | 2400 | 61 | 100 | 126–150 | 87.5$\pm$0.1 | 105.9$\pm$0.1 | 0.2
2400 | 3300 | 61 | 100 | 147–162 | 92.2$\pm$0.1 | 107.6$\pm$0.1 | 0.1
3300 | 4360 | 61 | 100 | 81–88 | 95.5$\pm$0.1 | 105.5$\pm$0.1 | 0.2
1365 | 4360 | 61 | 100 | 369–400 | 91.5$\pm$0.1 | 106.7$\pm$0.1 | 0.1
1365 | 2400 | 101 | 150 | 122–131 | 88.9$\pm$0.1 | 106.1$\pm$0.1 | 0.2
2400 | 3300 | 101 | 150 | 38–115 | 93.1$\pm$0.1 | 105.2$\pm$0.1 | 0.3
3300 | 4360 | 101 | 150 | 16–32 | 95.1$\pm$0.2 | 104.0$\pm$0.2 | 0.6
1365 | 4360 | 101 | 150 | 183–278 | 90.4$\pm$0.1 | 105.5$\pm$0.1 | 0.2
1365 | 2400 | 2 | 150 | 443–566 | 88.6$\pm$0.1 | 105.9$\pm$0.1 | 0.1
2400 | 3300 | 2 | 150 | 501–614 | 91.9$\pm$0.1 | 109.5$\pm$0.1 | 0.1
3300 | 4360 | 2 | 150 | 371–398 | 93.5$\pm$0.1 | 106.3$\pm$0.1 | 0.1
1365 | 4360 | 2 | 150 | 1381–1570 | 91.2$\pm$0.1 | 107.6$\pm$0.1 | 0.1

###  4.2 Low harmonic degrees

To emulate the observations of disk-integrated helioseismic observations, we
also averaged the lowest available harmonic degrees $l=2\text{--}4$ over the
complete frequency range. Figure 6 shows the resulting normalised variation as
a function of time. Due to the smaller number of azimuthal orders per
multiplet and the smaller number of multiplets overall (between 28 and 43,
depending on the time sample), the averages are less well constrained than in
the wider harmonic degree ranges shown in Figure 5. The solid red line shows
the running one-year variance-weighted mean with its uncertainty in the red
shaded band. As in Fig. 5 the error bars indicate $5\sigma$ uncertainties.

The extrema of these smoothed data are $92.2\pm 0.3\text{\,}\mathrm{\char
37\relax}$ and $106.4\pm 0.3\text{\,}\mathrm{\char 37\relax}$, where the
median uncertainty on the un-smoothed data point is $0.9\text{\,}\mathrm{\char
37\relax}$. We will discuss this further in Section 5.

Figure 6: Normalised variation of the energy supply rate of modes of harmonic
degrees $l=2\text{--}4$ as a function of time. Colour coding as in Figure 5.

###  4.3 Cross-correlating magnetic activity and $\frac{dE}{dt}$

To assess the correlation between the variation observed in $dE/dt$ and the
level of solar magnetic activity, as measured by the $\overline{F_{10.7}}$
index, we computed the cross-correlation function (CCF) for the independent
time samples, i.e., every third time step. We again split the modes into the
subsets we used before: the modes included in the calculation of the functions
shown in the panels of Figure 7 correspond to those of Figure 5. In Figure 7,
the cross-correlation functions are shown by solid blue lines with blue dots.
The first of the three possible sets of independent data points was chosen for
analysis. Fourth-order polynomial fits to the CCFs are shown by the solid red
curves. The dotted vertical blue line indicates the global minimum of each
panel’s CCF, while the dashed vertical red line indicates the minimum of the
fitted function. These fits are performed with numpy’s least-square polynomial
fit routine numpy.polyfit (Oliphant, 2006). The blue shaded bands indicate the
minimum-to-maximum range of the three possible independent data sets. The
correlation values typically vary only by a few percent between the three data
sets.

In Table 4 the Pearson correlation coefficient $r$ and its $p$-value for the
different mode sets are given for different lag values: for un-lagged data in
columns 5 and 6; for the lag of the minimum of the cross-correlation function,
in columns 7–9; for the lag of the minimum of the quadratic fit to the cross-
correlation function, in columns 10–12. The last two columns give Spearman’s
rank correlation $\rho$ for un-lagged data and its corresponding $p$-value.

For the un-lagged data the $p$-values are $<0.05$ for 15 out of the 20 mode
sets. For low harmonic degrees $l=2\text{--}30$ only the low frequency modes
have $p<0.05$. In addition, for the high frequency modes, the ranges
$l=61\text{--}100$ and $l=101\text{--}150$ have $p>0.05$.

This essentially also holds true for the other lag values (global minimum of
the CCF and minimum of the fitted function) and the Spearman $\rho$: For the
low-, and mid-frequency sets, as well as for the full mode set, the $p$-value
of the correlation coefficients is $<0.05$ for all harmonic degree ranges.
Indeed, overall the correlation coefficients indicate a moderate to strong
anti-correlation between the level of solar magnetic activity and the p-mode
energy supply rates.

The strongest anti-correlation with $r=-0.82$ is found for the low frequency
mode set with $l=101\text{--}150$ at zero lag. Except for the high harmonic
degree sets, the minima of all (statistically significant) CCFs are found at
positive lag values.

Figure 7: Cross-correlation functions (CCF) between the data in the corresponding panels of Fig. 5 and the $\overline{F_{10.7}}$ solar radio flux of Fig. 1 (blue curves and blue circles). The minimum value of each panel’s CCF is indicated by a vertical blue dotted line. Red curves show fourth-order polynomial fits to the CCF in the presented range of lags. The minimum value of each panel’s fit is indicated by a vertical red dashed line. The ordinate extends from -0.84 to 0.3 in all panels. The abscissa, i.e., the lag, is in units of GONG months, where one GONG month equals 36 . The blue shaded bands show the minimum-to-maximum range of the three independent sets of data of which the first set was chosen for the fit. Table 4: Correlation coefficients between mode energy supply rates of different parameter ranges (defined in columns 1–4) and the $\overline{F_{10.7}}$ solar radio flux for different lags: Pearson $r$ and corresponding $p$-value for un-lagged data in columns 5 and 6. Lag of the minimum of the cross-correlation function, this lag’s Pearson $r$ and $p$-value in columns 7–9. The lag of the minimum of the polynomial fit to the cross-correlation function, this lag’s Pearson $r$ and $p$-value in columns 10–12. Spearman’s rank correlation $\rho$ for un-lagged data and corresponding $p$-value in columns 13 and 14.  Frequency range [$\mathrm{\SIUnitSymbolMicro Hz}$] | Harmonic degrees | $r(0)$ | $p(0)$ | Lag${}_{\text{min}}$ [$\mathrm{d}$] | $r_{\text{min}}$ | $p_{\text{min}}$ | Lag${}_{\text{fit}}$ [$\mathrm{d}$] | $r_{\text{fit}}$ | $p_{\text{fit}}$ | $\rho$ | p
---|---|---|---|---|---|---|---|---|---|---|---
$\nu_{\text{min}}$ | $\nu_{\text{max}}$ | $l_{\text{min}}$ | $l_{\text{max}}$ |  |  |  |  |  |  |  |  |  |
1365 | 2400 | 2 | 30 | -0.43 | $<10^{-3}$ | 216 | -0.51 | $<10^{-3}$ | 540 | -0.51 | $<10^{-3}$ | -0.44 | $<10^{-3}$
2400 | 3300 | 2 | 30 | -0.08 | 0.49 | 864 | -0.27 | 0.019 | 864 | -0.27 | 0.019 | -0.02 | 0.84
3300 | 4360 | 2 | 30 | -0.15 | 0.17 | 756 | -0.33 | 0.0029 | 756 | -0.33 | 0.0029 | -0.07 | 0.51
1365 | 4360 | 2 | 30 | -0.18 | 0.1 | 756 | -0.34 | 0.0029 | 756 | -0.34 | 0.0029 | -0.12 | 0.27
1365 | 2400 | 31 | 60 | -0.48 | $<10^{-3}$ | 108 | -0.51 | $<10^{-3}$ | 432 | -0.47 | $<10^{-3}$ | -0.50 | $<10^{-3}$
2400 | 3300 | 31 | 60 | -0.29 | 0.0082 | 756 | -0.31 | 0.0053 | 432 | -0.30 | 0.006 | -0.24 | 0.025
3300 | 4360 | 31 | 60 | -0.46 | $<10^{-3}$ | 108 | -0.48 | $<10^{-3}$ | 432 | -0.48 | $<10^{-3}$ | -0.37 | $<10^{-3}$
1365 | 4360 | 31 | 60 | -0.39 | $<10^{-3}$ | 108 | -0.42 | $<10^{-3}$ | 432 | -0.40 | $<10^{-3}$ | -0.34 | 0.0013
1365 | 2400 | 61 | 100 | -0.75 | $<10^{-3}$ | 108 | -0.76 | $<10^{-3}$ | 324 | -0.71 | $<10^{-3}$ | -0.73 | $<10^{-3}$
2400 | 3300 | 61 | 100 | -0.42 | $<10^{-3}$ | 108 | -0.44 | $<10^{-3}$ | 432 | -0.41 | $<10^{-3}$ | -0.31 | 0.0037
3300 | 4360 | 61 | 100 | 0.03 | 0.8 | 864 | -0.22 | 0.061 | 1080 | -0.21 | 0.072 | 0.06 | 0.61
1365 | 4360 | 61 | 100 | -0.52 | $<10^{-3}$ | 108 | -0.54 | $<10^{-3}$ | 432 | -0.51 | $<10^{-3}$ | -0.44 | $<10^{-3}$
1365 | 2400 | 101 | 150 | -0.82 | $<10^{-3}$ | 0 | -0.82 | $<10^{-3}$ | 108 | -0.81 | $<10^{-3}$ | -0.79 | $<10^{-3}$
2400 | 3300 | 101 | 150 | -0.57 | $<10^{-3}$ | 0 | -0.57 | $<10^{-3}$ | 108 | -0.57 | $<10^{-3}$ | -0.48 | $<10^{-3}$
3300 | 4360 | 101 | 150 | -0.10 | 0.36 | -540 | -0.18 | 0.1 | -756 | -0.17 | 0.1 | -0.05 | 0.67
1365 | 4360 | 101 | 150 | -0.77 | $<10^{-3}$ | 0 | -0.77 | $<10^{-3}$ | 108 | -0.76 | $<10^{-3}$ | -0.70 | $<10^{-3}$
1365 | 2400 | 2 | 150 | -0.74 | $<10^{-3}$ | 108 | -0.75 | $<10^{-3}$ | 324 | -0.69 | $<10^{-3}$ | -0.72 | $<10^{-3}$
2400 | 3300 | 2 | 150 | -0.35 | $<10^{-3}$ | 540 | -0.38 | $<10^{-3}$ | 432 | -0.37 | $<10^{-3}$ | -0.25 | 0.022
3300 | 4360 | 2 | 150 | -0.30 | 0.0064 | 756 | -0.39 | $<10^{-3}$ | 540 | -0.38 | $<10^{-3}$ | -0.20 | 0.072
1365 | 4360 | 2 | 150 | -0.50 | $<10^{-3}$ | 108 | -0.52 | $<10^{-3}$ | 432 | -0.50 | $<10^{-3}$ | -0.43 | $<10^{-3}$

###  4.4 Comparison of activity extrema

We compare the energy supply rates during the different activity extrema
listed in Table 1 with each other. For this, we average the energy supply
rates of each mode over the time samples during the periods of Table 1, take
the difference between two extremal periods, and normalise by the mean over
the entire time series. As activity and energy supply rates are anti-
correlated, we subtract successive periods of high activity from periods of
low activity for better comparability of the results. Also, to assess the
relative depth or strength of minima and maxima, we compare the minima of
cycles 25 and 24 as well as the maxima of cycles 24 and 23. Only energy supply
rates of modes which are present in two periods can be subtracted from each
other. Thus, if two rows of Table LABEL:table:extr_change_freq_org are
subtracted from each other, e.g. row three from row two to seemingly yield
Max24-Max23, there is a small difference to the values given in the last row.
In this example, the last row only necessitates the modes to be present in
Max24 and Max23, whereas taking the difference between rows three and two
demands the modes to be present in all of these rows’ four extrema.

In Figure LABEL:fig:fracnu_unlagged, the normalised differences between the
different extrema are shown for all modes as a function of mode frequency
(black data points). The extrema that are compared in the respective panel are
indicated to the left of each panel. In order to smooth out the scatter in the
normalised differences, we calculated the rolling variance-weighted average
over 100 modes consecutive in mode frequency. In each panel, this is shown by
the solid coloured curve and its $10\sigma$ confidence interval is given by
the coloured band. The $1\sigma$ uncertainty is calculated as the standard
error of the weighted mean. Due to the heavy averaging that has been applied
up to this point, the uncertainties are rather small. Figure
LABEL:fig:fracell_unlagged shows the same as Figure LABEL:fig:fracnu_unlagged
but as a function of harmonic degree. Here, the weighted averaging is first
done over the modes of each individual degree and then the rolling variance-
weighted average over five consecutive degrees is calculated.

For better comparability, Figure 8 collects the smoothed curves from Figures
LABEL:fig:fracnu_unlagged and LABEL:fig:fracell_unlagged. Here, the left panel
shows the normalised differences as a function of mode frequency and the right
panel as a function of harmonic degree.

In Tables LABEL:table:extr_change_freq_org and
LABEL:table:extr_change_harmdeg_org, the percental energy supply rate change
of modes in the four mode frequency ranges and five harmonic degree ranges are
listed. We will discuss these results further in the next section.

##  5 Discussion

###  5.1 Variation of supply rates

#### Dependence on mode frequency and harmonic degree

The energy supply rates of solar p modes vary over the solar activity cycle.
The magnitude of this variation and the correlation with the
$\overline{F_{10.7}}$ index varies with mode frequency and harmonic degree.

From Fig. 5 and Table 3 we find that energy supply rates of modes in the range
$l=61\text{--}100$ with frequencies $<$2400\text{\,}\mathrm{\SIUnitSymbolMicro
Hz}$$ vary most strongly with a minimum of $87.5\pm 0.1\text{\,}\mathrm{\char
37\relax}$ during times of high activity and a maximum of $105.9\pm
0.1\text{\,}\mathrm{\char 37\relax}$ at low activity levels. The anti-
correlation of this mode set with activity is strong with $r(0)=-0.75$ and
$p<10^{-3}$. The strongest correlation is found for the mid-frequency mode set
with $l=101\text{--}150$, for which $r(0)=-0.82$ and $p<10^{-3}$.

The amplitude of the variation over the activity cycle is smaller for higher
frequency modes, which can be appreciated from Fig. 5, Table 3, and
particularly from Fig. 8. The anti-correlation at zero lag between these high
frequency modes and $\overline{F_{10.7}}$ has $p<0.05$ only for the range
$l=31\text{--}60$ and the full range of harmonic degrees.

Interestingly, the strongest – and most significant – anti-correlation values
for all mode sets except the highest harmonic degrees $l=101\text{--}150$ are
found at positive lag values. I.e., it takes the energy supply rates a certain
amount of time to react to changes in the level of magnetic activity.

#### Why has this not been detected before?

As the detected variation of $dE/dt$ over the activity cycle is rather large
and very significant for the majority of mode sets, it is worthwhile to ask
why this variation has not been detected until now.

Previous studies of the solar-cycle variation of p-mode parameters often
focused on mode frequencies (see Section 1 for references), as these hold the
most accessible information about the deep layers of the Sun. When studies did
include the other mode parameters – damping width $\Gamma$, mode amplitude
$A$, and quantities compound out of these two – they were almost exclusively
limited to harmonic degrees $l\leq 2$. The total number of azimuthal orders
per radial order of the multiplets $l=0,1,2$ is 9. What is more, this
effectively reduces to 6, as only modes for which $l+m$ is even are clearly
seen in Sun-as-a-star data. Thus, even in the case if all harmonic azimuthal
orders are fitted equally precisely (which is not the case for unresolved
observations e.g., Chaplin et al., 2004), this is equivalent to using only the
$l=3$ modes from our study and averaging over their azimuthal orders. Mind
that Figure 6 in fact includes $l=2\text{--}4$ from resolved observations,
i.e., 21 azimuthal orders per radial order.

In addition, it can be seen from the right panel of Figure 8 that lower
harmonic degrees have smaller normalised differences between activity extrema.
Thus, these modes’ $dE/dt$ appear to be less sensitive to activity compared to
modes of higher degree. The larger deviations from unity around the end of
Max${}_{\text{23}}$ and just before the beginning of Min24 seen in Figure 6
can also be found in the larger mode sets in Figure 5: low-degree modes do
appear to have at least some response to the changing level of activity.
Overall, the Pearson correlation-coefficient $r$ at zero lag between the
normalised differences of the $l=2\text{--}4$ mode set plotted in Figure 6 and
the $\overline{F_{10.7}}$ index is only $r(0)=-0.18$ with a $p$-value of 0.1.

From this we conclude that some combination of a smaller number of multiplets
in total, the small number of azimuthal orders to average over, the shorter
length of the observations, and the fact that lower harmonic degrees appear to
be less sensitive to magnetic activity, lead to the null results of earlier
studies on the matter of activity-related variations of $dE/dt$.

#### Sensitivity of $dE/dt$ to magnetic activity

Figure 9 shows the normalised variation of $dE/dt$ for the full mode set
(average over all frequencies and harmonic degrees) as a function of
$\overline{F_{10.7}}$. From a linear fit (blue line) to this we find that

$\displaystyle\left[dE/dt\right]=\left(1.050\pm 0.007\right)-\left(4.8\pm
0.6\right)\cdot
10^{-4}$\mathrm{s}\mathrm{f}\mathrm{u}^{-1}$\overline{F_{10.7}},$ (6)

where the square brackets indicate that the energy supply rates are normalised
to the mean over the full cycle and the entire mode set is included. Again,
the linear fit shows the anti-correlation of the energy supply rates with
activity: The quiet Sun supply rates are $>1$ and the slope is $<0$.

Figure 8: Smoothed differences between the energy supply rates of the extrema
of magnetic activity normalised by the mean over the entire GONG time series
as shown in Figs. LABEL:fig:fracnu_unlagged and LABEL:fig:fracell_unlagged.
Colours indicate the pairs of activity extrema that are subtracted from each
other as given at the top. Figure 9: Normalised variation of the full mode
set, which includes all frequencies and harmonic degrees, as a function of the
$\overline{F_{10.7}}$ index. The solid blue line is a linear regression of the
data.

Many of the data points above the linear fit belong to the rising activity
phase going from minimum 23 to maximum 23. Whether this is part of a
hysteresis pattern in $dE/dt$ over a full magnetic cycle or an artefact from
the GONG data and the GONG hardware upgrade just at the end of the maximum of
cycle 23 should be tested with data from other instruments. The much larger
percental change in $dE/dt$ between the minimum and maximum of cycle 23
compared to the later extrema can also be seen in Tables
LABEL:table:extr_change_freq_org and LABEL:table:extr_change_harmdeg_org. When
comparing Min24 to Min23 directly, we find that the supply rates of the full
mode set are $5.51\pm 0.02\text{\,}\mathrm{\char 37\relax}$ lower for Min24,
which is obviously in contrast to the surface tracers of activity, e.g., the
sunspot cycle. It is also conceivable that, due to the different strength of
cycle 24, the energy supply rates have settled at a slightly different level
when compared to the earlier stronger cycles which had larger amplitudes
between their activity minima and maxima.

As Figure LABEL:fig:fracnu_unlagged and Table LABEL:table:extr_change_freq_org
show, modes around and above the Sun’s frequency of maximum oscillation
amplitude $\nu_{\text{max}}$ ($\approx$3080\text{\,}\mathrm{\SIUnitSymbolMicro
Hz}$$, e.g., Kiefer et al. 2018) are less sensitive to magnetic activity than
modes below $\nu_{\text{max}}$. Interestingly, this is in contrast to mode
frequencies for which the amplitude of change along the cycle increases with
mode frequency (e.g., Broomhall, 2017). Furthermore, let us consider the
results of Kiefer et al. (2018), in particular their figures 6 and 7 which
show the analogons of our Figure 5 but for mode widths $\Gamma$ and mode
amplitudes $A$. While Kiefer et al. (2018) did not compare activity extrema
separately, it can be appreciated that the percental variation in the low
frequency mode set going from one activity extremum to the next is larger for
$dE/dt$ than it is for $\Gamma$ or $A$. Also, the variation is larger for the
low frequency mode set than the mid frequency set for $dE/dt$, but for
$\Gamma$ and $A$ separately, the variation is larger in the mid frequency set.
This is most apparent for the damping widths $\Gamma$ in figure 6 of Kiefer et
al. (2018). All this shows that it is indeed not an increase in the damping
alone that is causing the variations in $\Gamma$ and $A$ as has been assumed
thus far. Indeed these observations call for an activity related variation in
the forcing function of solar p modes.

#### Lag between $dE/dt$ and $\overline{F_{10.7}}$

As we have established, the high frequency mode set is least reliable for the
purpose of investigating $dE/dt$; so let us exclude this set from the
considerations of this subsection and focus on the other three frequency
ranges.

Looking at the minima of the CCFs between $dE/dt$ and $\overline{F_{10.7}}$
and the minima of the fits to them, we find that the lag value for these
decreases with increasing harmonic degree. For example, in the mid frequency
range, the position of the global minimum of the CCF decreases from
$864\text{\,}\mathrm{d}$ for modes in the range $l=2\text{--}30$, to
$756\text{\,}\mathrm{d}$ for $l=31\text{--}60$, to $108\text{\,}\mathrm{d}$
for $l=61\text{--}100$, to finally $0\text{\,}\mathrm{d}$ for modes with
$l=101\text{--}150$. This is interesting as the higher harmonic degree modes
are confined to more shallow layers than modes of lower harmonic degrees
(Basu, 2016). It thus appears that surface activity affects those modes
earlier which are confined closer to the surface and modes that penetrate
deeper into the Sun are fully affected later. As p modes are assumed to be
excited very close to the surface (e.g., Chaplin & Appourchaux, 1999) it is
not immediately apparent why low degree, high inertia modes should take longer
to react to changes in surface activity. We speculate that the higher mode
inertia of lower frequency and lower harmonic degree modes acts as a sort of
buffer against changes in the mode forcing, i.e., a stronger or longer-lasting
perturbation is needed to induce a change in their energy supply rates.

We further note the curious behaviour in all mode sets’ CCF just after the
zero lag: compared to the polynomial fit, the CCF dips to smaller correlation
values at around a lag of $108\text{\,}\mathrm{d}$. It appears to oscillate
back above the fit only to decrease again slightly. We speculate that this
(quasi-)oscillatory behaviour in the CCF might be connected to the Sun’s
quasi-biennial variations that have been observed in, e.g., p-mode frequencies
(Broomhall et al., 2012). This dip persists even if all time samples are
included in the calculation of the cross-correlation functions. This feature
is thus likely not caused by the time resolution of $108\text{\,}\mathrm{d}$
used in Fig. 7.

#### Quasi-yearly variation of $\Theta$

Even after removing the residual change with apparent solar radius, the energy
supply parameter $\Theta$ in Figure 2 shows a quasi-yearly variation that
needs to be further investigated in future studies. We attempted to remove
this variation by fitting $\Theta_{nl}$ as a function of the solar position
angle $P$ and the $B_{0}$ angle. However, whatever small variation with these
quantities were removed by the linear fit, the observed variation in $\Theta$
persisted. We thus refrained from considering this matter further and call for
a more detailed investigation of this (quasi-)yearly variation. We have not
investigated periodograms of the variation of $dE/dt$. It is therefore
entirely possible that the short term variation that can so clearly be seen in
Figure 2 has a period that is quite different from
$1\text{\,}\mathrm{y}\mathrm{r}$ and might be connected to the above mentioned
quasi-biennal oscillations. Part of the residual seasonal variation may be
connected to changes in the GONG leakage matrix (Hill & Howe, 1998).

###  5.2 Impact on asteroseismology

The accurate prediction of the amplitudes of stellar p-mode oscillations is
crucial in the target selection of photometric space missions. Such a target
selection has been done for the asteroseismic targets of the NASA Transiting
Exoplanet Survey Satellite mission (TESS, Ricker et al., 2014) based on the
results of Campante et al. (2016) who calculated a detection probability for
solar-like oscillations. Their equation for the prediction of the oscillation
amplitudes in the power spectrum contains factors that account for the length
of the data strings, sampling and instrumental effects, the pattern of p-mode
peaks in the power spectrum and a factor that gives the maximum oscillation
amplitude of radial modes. The calculation of this amplitude factor is based
on scaling relations based on the work of Kjeldsen & Bedding (1995) and
Chaplin et al. (2011). Campante et al. (2016) mention the need for accounting
for the effect of magnetic activity in the estimation of mode amplitudes.

During the Kepler mission (Koch et al., 2010; Borucki et al., 2010), over 2000
stars were observed in the survey phase. Of these, only 540 were detected to
exhibit solar-like oscillations (Mathur et al., 2019). It is known that – as
in the solar case – stellar oscillations are suppressed by magnetic activity
(e.g., García et al., 2010; Salabert et al., 2017; Kiefer et al., 2017; Santos
et al., 2018). Indeed, Mathur et al. (2019) were able to attribute
$32\text{\,}\mathrm{\char 37\relax}$ of non-detections to magnetic activity.
The remaining $68\text{\,}\mathrm{\char 37\relax}$ however are still to be
accounted for.

Based on the results we present in this article, we conjecture that it is not
only increased damping of oscillations and the ensuing attrition of mode
amplitudes that impede the detection of stellar p-modes and need to be
accounted for, but in fact the oscillations are fed with less energy per unit
time even for moderately active stars like the Sun.

In the target selection of future asteroseismic surveys this should be taken
into account for stars where a reliable measure of activity is available. If
the detection probability of oscillations is to be maximised, more active
stars ought to be deprioritised. More work on this has to be done, e.g., on
the extent to which the lowest harmonic degrees are affected by activity. This
problem could be tackled by careful analyses of the available photometric time
series on the one hand (Salabert et al., 2017) and simulations of mode
excitation in solar and stellar convection zones (e.g., Belkacem et al., 2006;
Samadi et al., 2007; Zhou et al., 2019; Zhou et al., 2020) that investigate
the dependence of the mode excitation rate in these simulations on the
magnetic field strength on the other hand.

###  5.3 Shortcomings

We have corrected the jumps in mode amplitudes that were noticed by Kiefer et
al. (2018) in the same empirical way they used: we applied a correction factor
to remove the two obvious jumps (see Section 3.1). This is unsatisfactory, as
neither the origin of the second jump is explained, nor can we be certain that
the direction of the adjustment is correct. It is entirely possible that the
absolute values of the mode energy supply rates as presented in Figures 3, 4
and Table 5 are underestimated, as the correction factor assumes the time
between 2001 and 2005 as the baseline. Data before the jump in 2001 are
corrected downwards, data after the 2005 jump are corrected upwards. All our
time dependent analyses are unaffected by this choice of a baseline, only
statements about the absolute values of $dE/dt$ are concerned with this.

We have neglected possible contributions from time-dependent mode asymmetry in
this study. We are aware that some fraction of the detected activity-related
change of supply rates can potentially be attributed to changes in mode
asymmetry. With the publicly available official GONG data such analyses cannot
be done, as the profiles that are used in the GONG mode fitting pipeline are
symmetrical. It is therefore expedient to repeat similar analyses as we have
done here with data that include a mode asymmetry parameter (e.g., Korzennik,
2017) and, in particular, with data from the SOHO-MDI (Domingo et al., 1995;
Scherrer et al., 1995) and SDO-MDI (Pesnell et al., 2012; Schou et al., 2012)
instruments.

The GONG leakage matrix is not included in the fitting of the mode peaks. We
account for part of this shortcoming by removing the bowl-shape of the energy
supply rates as a function of $|m/l|$, see Section 3.1. This effectively takes
care of the $m$-dependence of the self-leakage. However, the overall level of
self-leakage is only naively corrected for by the inclusion on the factor
$C_{\text{vis}}$ in Eq. 1. We stress that using this factor for all harmonic
degrees is consistent with Komm et al. (2000b). In reality however, it can
vary for different harmonic degrees (see, e.g., Baudin et al., 2005). We do
not anticipate $C_{\text{vis}}$ to vary over the course of the Sun’s activity
cycle, thus it cannot contribute to the detected cycle variation – it only
affects the absolute values of energy supply rates. In a numerical estimation
of the geometrical visibilities of different harmonic degrees in resolved
Doppler observations, we found that the self-leakage levels off at relatively
low harmonic degree $l\approx 4$. Thus, the reported absolute averages of
energy supply rates are most likely only off by a small constant factor, as
the majority of modes have $l>4$.

##  6 Summary and Conclusions

We have, for the first time, analysed the solar p-mode energy supply rates up
to $l=150$ through two solar activity cycles. Contrary to the long-held
opinion that the energy that is fed into the p-modes per unit time is
constant, we have found that they are indeed sensitive to the level of
magnetic activity.

Averaged over the full mode set, the minimum supply rates are found during
activity maxima, with a minimum of $91.2\pm 0.1\text{\,}\mathrm{\char
37\relax}$ normalised to the entire time series. At activity minimum they
reach as high as $107.6\pm 0.1\text{\,}\mathrm{\char 37\relax}$, i.e., supply
rates are in anti-phase with the activity cycle with a correlation of
$r=-0.50$ $(p<10^{-3})$, where the $F_{10.7}$ index was used as a proxy for
solar activity.

The fractional change through the activity cycle depends somewhat on mode
frequency and only weakly on harmonic degree: supply rates of lower frequency
modes $<$3000\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$$ are more strongly
affected than higher frequency modes. It appears that the supply rates of
lower harmonic degree modes $l\lesssim 20$ tend to be least affected by
activity.

According to the mode energy supply rates, the currently ongoing minimum is as
deep as the long minimum between cycles 23 and 24. Averaged over the full mode
set, there is very little variation between these two minima with a difference
of only $0.33\pm 0.02\text{\,}\mathrm{\char 37\relax}$. Also, it is reflected
in the energy supply rates that the maximum of cycle 23 was stronger than that
of cycle 24: the energy supply rates were $2.43\pm 0.02\text{\,}\mathrm{\char
37\relax}$ higher during the latter.

As the energy supply rates are a crucial ingredient in simulations and
prediction of the solar and stellar p-mode amplitudes, it is worthwhile to
establish their ground-state, i.e., quiet Sun levels. We provide a table with
the energy supply rates as a function of mode frequency for the average over
the entire time series, corrected for mode inertia, averaged over the periods
of activity minima as well as maxima, and with the mode inertiae normalised at
both the photospheric radius $R_{\odot}$ as well as the GONG observation
height $r_{\text{obs}}$. Table 5 as well as a larger table with the energy
supply rates of 1606 modes are provided online.

Our findings could – and should – be corroborated by analysing the last two
activity cycles with data from multiple instruments. Optimally, such a study
is done with results obtained with both symmetrical and asymmetrical mode
profiles. This way, it could be determined to what extent the here reported
change in supply rates over the cycle is in fact due to variations in mode
asymmetry. Finally, we note that the convective forcing of acoustic mode in
stars with different metallicity and surface gravity than the Sun might
respond more weakly or strongly to magnetic activity.

## Acknowledgements

R. K. and A-M. B. acknowledge the support of the Science and Technology
Facilities Council (STFC) consolidated grant ST/P000320/1. A-M. B.
acknowledges the support STFC consolidated grant ST/T000252/1.

We thank Rudi Komm for supplying us with his code underlying Komm et al.
(2000b), which we used as a starting point for this work. The authors thank
the anonymous referee for taking the time to review this article and for their
useful comments.

This work utilises GONG data obtained by the NSO Integrated Synoptic Program
(NISP), managed by the National Solar Observatory, the Association of
Universities for Research in Astronomy (AURA), Inc. under a cooperative
agreement with the National Science Foundation. The data were acquired by
instruments operated by the Big Bear Solar Observatory, High Altitude
Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto
de Astrofísica de Canarias, and Cerro Tololo Interamerican Observatory.

software: This work made use of the following software libraries not all of
which is cited in the text: matplotlib (Hunter, 2007), NumPy (Oliphant, 2006),
pandas (Reback et al., 2019), SciPy (Virtanen et al., 2020), seaborn (Waskom
et al., 2018), statsmodels (Seabold & Perktold, 2010), SunPy v1.1.2 (Mumford
et al., 2020), an open-source and free community-developed solar data analysis
Python package (The SunPy Community et al., 2020), uncertainties (Lebigot,
2019).

## Data Availability

The data underlying this article are available from the GONG FTP server, at
ftp://gong2.nso.edu/TSERIES/v1f and from the Natural Resources Canada FTP
server
ftp://ftp.seismo.nrcan.gc.ca/spaceweather/solar_flux/daily_flux_values/.

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##  Appendix A Table of energy supply rates

Table 5: Energy supply rates averaged over 32 modes consecutive in mode frequency; the last row only contains 12 modes. The first column gives the mean mode frequency of each bin. The second and third columns are energy supply rates without and with rescaling by $Q$ averaged over the entire time series with variance-weighting over the 32 modes. The values in the following columns are averaged over only the indicated periods of activity extrema, i.e., all activity minima or maxima as defined in Table 1, and are scaled by $Q$ as indicated. To maintain readability, uncertainties are rounded up to 0.001. More digits are given in the online version of this table. Mode Frequency | $\dfrac{dE}{dt}$ | $\dfrac{dE}{dt}\cdot Q$ | $\dfrac{dE}{dt}\left(\text{Min}\right)$ | $\dfrac{dE}{dt}\left(\text{Max}\right)$ | $\dfrac{dE}{dt}\cdot Q\left(\text{Min}\right)$ |  $\dfrac{dE}{dt}\cdot Q\left(\text{Max}\right)$
---|---|---|---|---|---|---
$\left[$\mathrm{\SIUnitSymbolMicro Hz}$\right]$ | $\left[${10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$\right]$ | $\left[${10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$\right]$ | $\left[${10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$\right]$ | $\left[${10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$\right]$ | $\left[${10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$\right]$ |  $\left[${10}^{22}\text{\,}\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$\right]$
1449.29 | 0.105$\pm$0.001 | 0.037$\pm$0.001 | 0.109$\pm$0.001 | 0.106$\pm$0.001 | 0.038$\pm$0.001 | 0.037$\pm$0.001
1548.62 | 0.139$\pm$0.001 | 0.055$\pm$0.001 | 0.144$\pm$0.001 | 0.140$\pm$0.001 | 0.056$\pm$0.001 | 0.054$\pm$0.001
1616.52 | 0.177$\pm$0.001 | 0.074$\pm$0.001 | 0.183$\pm$0.001 | 0.177$\pm$0.001 | 0.076$\pm$0.001 | 0.073$\pm$0.001
1673.87 | 0.215$\pm$0.001 | 0.094$\pm$0.001 | 0.223$\pm$0.001 | 0.214$\pm$0.001 | 0.097$\pm$0.001 | 0.094$\pm$0.001
1724.75 | 0.276$\pm$0.001 | 0.120$\pm$0.001 | 0.285$\pm$0.001 | 0.274$\pm$0.001 | 0.123$\pm$0.001 | 0.119$\pm$0.001
1775.14 | 0.289$\pm$0.001 | 0.149$\pm$0.001 | 0.299$\pm$0.001 | 0.289$\pm$0.001 | 0.154$\pm$0.001 | 0.149$\pm$0.001
1833.03 | 0.354$\pm$0.001 | 0.189$\pm$0.001 | 0.367$\pm$0.001 | 0.350$\pm$0.001 | 0.195$\pm$0.001 | 0.187$\pm$0.001
1889.43 | 0.430$\pm$0.001 | 0.242$\pm$0.001 | 0.446$\pm$0.001 | 0.425$\pm$0.001 | 0.250$\pm$0.001 | 0.240$\pm$0.001
1942.92 | 0.553$\pm$0.001 | 0.303$\pm$0.001 | 0.573$\pm$0.001 | 0.547$\pm$0.001 | 0.313$\pm$0.001 | 0.300$\pm$0.001
1994.13 | 0.657$\pm$0.001 | 0.376$\pm$0.001 | 0.677$\pm$0.001 | 0.649$\pm$0.001 | 0.388$\pm$0.001 | 0.372$\pm$0.001
2041.96 | 0.761$\pm$0.001 | 0.454$\pm$0.001 | 0.784$\pm$0.001 | 0.749$\pm$0.001 | 0.468$\pm$0.001 | 0.447$\pm$0.001
2089.88 | 0.894$\pm$0.001 | 0.542$\pm$0.001 | 0.923$\pm$0.001 | 0.878$\pm$0.001 | 0.559$\pm$0.001 | 0.533$\pm$0.001
2136.39 | 1.058$\pm$0.001 | 0.658$\pm$0.001 | 1.094$\pm$0.001 | 1.041$\pm$0.001 | 0.680$\pm$0.001 | 0.648$\pm$0.001
2188.75 | 1.214$\pm$0.001 | 0.799$\pm$0.001 | 1.254$\pm$0.001 | 1.190$\pm$0.001 | 0.824$\pm$0.001 | 0.783$\pm$0.001
2241.29 | 1.476$\pm$0.001 | 0.987$\pm$0.001 | 1.533$\pm$0.001 | 1.448$\pm$0.001 | 1.023$\pm$0.001 | 0.969$\pm$0.001
2290.98 | 1.742$\pm$0.001 | 1.177$\pm$0.001 | 1.807$\pm$0.001 | 1.717$\pm$0.001 | 1.219$\pm$0.001 | 1.160$\pm$0.001
2338.77 | 2.033$\pm$0.001 | 1.410$\pm$0.001 | 2.103$\pm$0.001 | 2.002$\pm$0.001 | 1.457$\pm$0.001 | 1.393$\pm$0.001
2384.46 | 2.357$\pm$0.001 | 1.657$\pm$0.001 | 2.447$\pm$0.001 | 2.323$\pm$0.002 | 1.716$\pm$0.001 | 1.637$\pm$0.001
2436.51 | 2.670$\pm$0.001 | 1.901$\pm$0.001 | 2.744$\pm$0.001 | 2.624$\pm$0.002 | 1.954$\pm$0.001 | 1.872$\pm$0.001
2488.52 | 3.085$\pm$0.001 | 2.221$\pm$0.001 | 3.185$\pm$0.002 | 3.016$\pm$0.002 | 2.286$\pm$0.001 | 2.178$\pm$0.001
2538.01 | 3.594$\pm$0.001 | 2.571$\pm$0.001 | 3.715$\pm$0.002 | 3.509$\pm$0.002 | 2.652$\pm$0.001 | 2.521$\pm$0.002
2586.67 | 4.060$\pm$0.001 | 2.867$\pm$0.001 | 4.190$\pm$0.002 | 3.977$\pm$0.003 | 2.956$\pm$0.002 | 2.820$\pm$0.002
2632.23 | 4.368$\pm$0.001 | 3.214$\pm$0.001 | 4.512$\pm$0.002 | 4.266$\pm$0.003 | 3.308$\pm$0.002 | 3.151$\pm$0.002
2676.88 | 4.814$\pm$0.001 | 3.580$\pm$0.001 | 4.950$\pm$0.003 | 4.718$\pm$0.004 | 3.681$\pm$0.002 | 3.524$\pm$0.003
2725.07 | 5.655$\pm$0.001 | 4.052$\pm$0.001 | 5.819$\pm$0.003 | 5.540$\pm$0.004 | 4.166$\pm$0.002 | 3.979$\pm$0.003
2775.65 | 5.817$\pm$0.002 | 4.458$\pm$0.001 | 6.000$\pm$0.003 | 5.701$\pm$0.004 | 4.585$\pm$0.002 | 4.382$\pm$0.003
2824.68 | 6.868$\pm$0.002 | 4.955$\pm$0.001 | 7.066$\pm$0.004 | 6.704$\pm$0.005 | 5.082$\pm$0.003 | 4.856$\pm$0.003
2872.46 | 7.090$\pm$0.002 | 5.467$\pm$0.002 | 7.257$\pm$0.004 | 6.945$\pm$0.005 | 5.588$\pm$0.003 | 5.367$\pm$0.004
2917.83 | 8.238$\pm$0.002 | 6.039$\pm$0.002 | 8.429$\pm$0.005 | 8.031$\pm$0.006 | 6.161$\pm$0.003 | 5.903$\pm$0.004
2961.81 | 8.913$\pm$0.003 | 6.632$\pm$0.002 | 9.060$\pm$0.005 | 8.740$\pm$0.006 | 6.730$\pm$0.004 | 6.506$\pm$0.004
3009.36 | 9.324$\pm$0.002 | 7.059$\pm$0.002 | 9.479$\pm$0.005 | 9.137$\pm$0.006 | 7.183$\pm$0.004 | 6.911$\pm$0.004
3058.12 | 10.167$\pm$0.003 | 7.703$\pm$0.002 | 10.243$\pm$0.005 | 9.967$\pm$0.006 | 7.765$\pm$0.004 | 7.558$\pm$0.005
3105.20 | 10.858$\pm$0.003 | 8.138$\pm$0.002 | 10.946$\pm$0.006 | 10.576$\pm$0.007 | 8.209$\pm$0.004 | 7.939$\pm$0.005
3151.63 | 11.412$\pm$0.003 | 8.669$\pm$0.002 | 11.565$\pm$0.006 | 11.083$\pm$0.008 | 8.760$\pm$0.004 | 8.458$\pm$0.006
3196.12 | 11.788$\pm$0.003 | 9.157$\pm$0.002 | 11.823$\pm$0.006 | 11.552$\pm$0.008 | 9.176$\pm$0.005 | 9.004$\pm$0.006
3239.26 | 12.651$\pm$0.003 | 9.548$\pm$0.002 | 12.742$\pm$0.006 | 12.366$\pm$0.008 | 9.593$\pm$0.005 | 9.372$\pm$0.006
3281.75 | 12.685$\pm$0.004 | 9.730$\pm$0.003 | 12.720$\pm$0.007 | 12.408$\pm$0.009 | 9.742$\pm$0.005 | 9.558$\pm$0.007
3326.58 | 12.837$\pm$0.003 | 10.046$\pm$0.003 | 12.854$\pm$0.006 | 12.632$\pm$0.008 | 10.042$\pm$0.005 | 9.898$\pm$0.006
3373.55 | 12.715$\pm$0.003 | 10.075$\pm$0.003 | 12.694$\pm$0.006 | 12.469$\pm$0.008 | 10.067$\pm$0.005 | 9.906$\pm$0.006
3418.65 | 13.117$\pm$0.004 | 10.370$\pm$0.003 | 13.053$\pm$0.007 | 12.808$\pm$0.009 | 10.326$\pm$0.006 | 10.164$\pm$0.007
3462.49 | 13.394$\pm$0.004 | 10.425$\pm$0.003 | 13.483$\pm$0.007 | 13.129$\pm$0.009 | 10.493$\pm$0.005 | 10.252$\pm$0.007
3506.96 | 12.910$\pm$0.004 | 10.452$\pm$0.003 | 12.942$\pm$0.007 | 12.620$\pm$0.009 | 10.487$\pm$0.006 | 10.267$\pm$0.007
3555.87 | 13.142$\pm$0.004 | 10.673$\pm$0.003 | 13.181$\pm$0.007 | 12.783$\pm$0.009 | 10.720$\pm$0.006 | 10.432$\pm$0.007
3607.76 | 13.309$\pm$0.004 | 10.939$\pm$0.003 | 13.540$\pm$0.008 | 13.019$\pm$0.009 | 11.123$\pm$0.006 | 10.737$\pm$0.008
3666.86 | 12.817$\pm$0.004 | 10.664$\pm$0.003 | 13.144$\pm$0.008 | 12.606$\pm$0.010 | 10.920$\pm$0.006 | 10.500$\pm$0.008
3736.66 | 11.349$\pm$0.004 | 9.596$\pm$0.003 | 11.687$\pm$0.008 | 11.181$\pm$0.010 | 9.872$\pm$0.007 | 9.460$\pm$0.008
3830.47 | 8.498$\pm$0.003 | 7.385$\pm$0.003 | 8.674$\pm$0.006 | 8.362$\pm$0.008 | 7.533$\pm$0.006 | 7.270$\pm$0.007
3957.56 | 6.383$\pm$0.003 | 5.785$\pm$0.002 | 6.522$\pm$0.005 | 6.317$\pm$0.006 | 5.906$\pm$0.005 | 5.726$\pm$0.006
4125.98 | 4.589$\pm$0.002 | 4.341$\pm$0.002 | 4.709$\pm$0.004 | 4.511$\pm$0.005 | 4.453$\pm$0.004 | 4.269$\pm$0.005
4282.77 | 2.882$\pm$0.003 | 2.765$\pm$0.003 | 2.945$\pm$0.005 | 2.856$\pm$0.007 | 2.825$\pm$0.005 | 2.739$\pm$0.006