11institutetext: School of Physics and Astronomy, University of Minnesota, 116
Church St. SE Minneapolis
11email<EMAIL_ADDRESS>22institutetext: Department of Physics and
Astronomy,University of Iowa, Iowa City, IA 52242, USA 33institutetext: Space
Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA
44institutetext: Climate and Space Sciences and Engineering, University of
Michigan, Ann Arbor, MI 48109, USA 55institutetext: Smithsonian Astrophysical
Observatory, Cambridge, MA 02138 USA 66institutetext: LPC2E, CNRS and
University of Orléans, Orléans, France 77institutetext: Solar System
Exploration Division, NASA/Goddard Space Flight Center, Greenbelt, MD, 20771
88institutetext: Laboratory for Atmospheric and Space Physics, University of
Colorado, Boulder, CO 80303, USA 99institutetext:  LESIA, Observatoire de
Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place
Jules Janssen, 92195 Meudon,France 1010institutetext: Department of Physics,
University of California, Berkeley, Berkeley, CA 94709 USA 1111institutetext:
Department of Astrophysical and Planetary Sciences, University of Colorado,
Boulder, CO, USA

# Narrowband oblique whistler-mode waves: Comparing properties observed by
Parker Solar Probe at ¡0.3 AU and STEREO at 1 AU

C. Cattell 11 B. Short 11 A. Breneman 11 J. Halekas 22 P. Whittesley 33 D.
Larson 33 J. Kasper 44 M. Stevens 55 T. Case 55 M. Moncuquet 99 S. Bale 331010
J. Bonnell 33 T. Dudok de Wit 66 K. Goetz 11 P. Harvey 33 R. MacDowall 77 D.
Malaspina 881111 M. Maksimovic 99 M. Pulupa 33 K. Goodrich 33

###### Abstract

Aims. Large amplitude narrowband obliquely propagating whistler-mode waves at
frequencies of $\sim$0.2 $f_{ce}$(electron cyclotron frequency) are commonly
observed at 1 AU, and are most consistent with the whistler heat flux fan
instability. We want to determine whether similar whistler-mode waves occur
inside 0.3 AU, and how their properties compare to those at 1 AU.

Methods. We utilize the waveform capture data from the Parker Solar Probe
Fields instrument from Encounters 1 through 4 to develop a data base of
narrowband whistler waves. The SWEAP instrument, in conjunction with the
quasi-thermal noise measurement from Fields, provides the electron heat flux,
beta, and other electron parameters.

Results. Parker Solar Probe observations inside $\sim$0.3 AU show that the
waves are often more intermittent than at 1 AU, and interspersed with
electrostatic whistler/Bernstein waves at higher frequencies. This is likely
due to the more variable solar wind observed closer to the Sun. The whistlers
usually occur within regions when the magnetic field is more variable and
often with small increases in the solar wind speed. The near-sun whistler-mode
waves are also narrowband and large amplitude, and associated with beta
greater than 1.The association with heat flux and beta is generally consistent
with the whistler fan instability. Strong scattering of strahl energy
electrons is seen in association with the waves, providing evidence that the
waves regulate the electron heat flux.

###### Key Words.:

Physical data and process:Instabilities, plasmas, waves,(Sun:) solar wind,Sun:
heliosphere

##  1 Introduction

Determining which wave modes control the evolution of solar wind electrons has
long been of interest, from the early studies of their properties,
characterizing three populations – core, halo and strahl (Feldman et al.,
1975). Observations indicated that the pitch angle width of strahl was much
broader at 1 AU than would be expected due to the conservation of the magnetic
moment. In addition to collisional scattering, various wave modes were
examined to see if they could provide the required scattering. Early
theoretical work was hampered by the lower time resolution measurements of
wave spectra obtained by spacecraft in the solar wind. The development of
waveform capture instruments provided high time resolution full waveform data.
Studies utilizing STEREO waveform data near 1 AU revealed the presence of
large amplitude, narrowband whistler-mode waves with frequencies of $\sim$0.2
$f_{ce}$. The waves propagate at highly oblique angles to the solar wind
magnetic field with significant parallel electric fields enabling strong
interaction with solar wind electrons without requiring the counter-
propagation needed with parallel propagating waves. These waves are frequently
observed, most often in association with stream interaction regions (SIRs),
but also within coronal mass ejections (CMEs) (Breneman et al., 2010; Cattell
et al., 2020a) and wave groups can be observed to last for intervals of days.

Inside $\sim$0.3 AU, Parker Solar Probe data indicate that electrostatic waves
at higher frequencies ($\sim$0.7 to several times $f_{ce}$) may be more common
(Malaspina et al., 2020), particularly in regions of quiet radial magnetic
field. These waves include both electron Bernstein and electrostatic whistler-
mode waves. The occurrence frequency decreases with distance from the Sun,
consistent with their absence in the STEREO waveform data at 1 AU. Lower
frequency sunward propagating whistler-mode waves are also observed by Parker
Solar Probe (Agapitov et al., 2020), primarily in association with decreases
in the magnetic field or the rapid change in magnetic field orientation called
‘switchbacks’ or jets (Bale et al., 2019; Kasper et al., 2019).

The properties of the electron distributions have been characterized inside
$\sim$.2 AU by Parker Solar Probe (Halekas et al., 2020a, b), between
$\sim$.3AU and$\sim$.75 AU by Helios, at 1 AU by Wind and Cluster, and outside
1 AU by Ulysses (Maksimovic et al., 2005; Štverák et al., 2009; Wilson III et
al., 2019). Although the radial dependence of the changes in the properties of
core, halo and strahl are consistent between these studies, the specific
mechanisms that provide the scattering and energization have not been
definitely identified. To understand the role the observed narrow-band
whistler-mode waves play in modifying the electron distributions and
regulating heat flux, it is important to determine how their occurrence and
properties depend on distance from the Sun.

In this report, we describe comparisons of narrowband whistler-mode waves
observed in the waveform data obtained by Parker Solar Probe from Encounters 1
through 4, and by STEREO. Section 2 presents the data sets and methodology.
Example waveforms and statistical results on the waves are discussed in
Section 3. Conclusions and possible consequences for solar wind evolution are
presented in Section 4.

##  2 Data sets and methodology

We utilize the Level 2 waveform capture data obtained during the first four
solar encounters by the Parker Solar Probe Fields Suite (Bale et al., 2016).
The details of the waveform capture instrument are described by Malaspina et
al. (2016). During the first encounter, three components of the magnetic field
using the search coil instrument were obtained, enabling determination of the
wave vector direction. Subsequent encounters obtained two components. Although
three components of the electric field (potential difference across probes)
are transmitted, we utilize primarily the two components in the plane
perpendicular to the spacecraft-Sun line obtained by the longer antennas. A
boom length of 3.5 m is used to covert potential differences to electric
fields; a smaller effective boom length would increase electric field
amplitudes. The waveform data utilized in this study were obtained for 3.5 s
intervals at 150 ksamples/s. As implemented on STEREO, the highest quality
(usually defined by amplitude of the electric field) captures are stored and
transmitted. In addition, intervals of interest in the summary data were
selected by the Fields team for transmission of waveform data to the ground.
Note that in the first three encounters dust impacts often triggered the
quality flag. For later encounters, software modifications reduced the number
of dust triggers. The wave amplitudes obtained from the first three encounters
are therefore, on average, smaller than those from the fourth. We also utilize
one electric field and one magnetic field channel in the DC coupled spectral
data, which is obtained at a rate of 1 spectra / 64 Cy, where 1 Cy = 0.873813
s, over a frequency range of $\sim$10 Hz to 4.8 kHz (Malaspina et al., 2016).
The spectra are $\sim$30 s averages. We have also examined one electric field
and one magnetic field channel in the DC coupled bandpass filter (BPF) data
which is obtained at a higher cadence of 1 spectrum/ 2 Cy.

The electron parameters were obtained from the Parker Solar Probe Solar Wind
Electrons Alphas and Protons Investigation (SWEAP) (Kasper et al., 2016) Solar
Probe Analyzers (SPAN-A-E and SPAN-B-E) (Whittlesey et al., 2020). We utilize
electron temperature, temperature anisotropy, heat flux and density moments
and the pitch angle distributions for energies from 2 to 2000 eV, covering
core, halo and strahl (Halekas et al., 2020a, b). The solar wind velocity was
obtained from the Level 2 Solar Probe Cup (SPC) moments (Case et al., 2020).
The solar wind density, and core and suprathermal electron temperatures were
obtained from the Fields Quasi-thermal Noise (QTN) data (Moncuquet et al.,
2020).

##  3 Waveform examples and statistics

Figure 1 presents an overview of a 31 hour interval from 12 UT on November 2,
2018 to 19 UT on November 3, 2018 that included 9 waveform captures with
narrowband whistlers, as well as higher frequency electrostatic waves. The top
two panels, which plot the DC-coupled BPF electric field spectrum and the DC-
coupled BBF magnetic field from 12 to 4000 Hz clearly shows the distinction
between the higher frequency electrostatic whistlers/Bernstein waves discussed
by Bale et al. (2019) and Malaspina et al. (2020) and the narrowband whistlers
that are the focus of this letter. Examples of the higher frequency
electrostatic waves are at $\sim$1615 to 1715 UT on November 2, and
intermittently between $\sim$03 and 05 UT on November 3, as well as for
shorter intervals on both days. Examples of the narrowband electromagnetic
whistlers can been seen in both spectra at $\sim$1700 to 1740 UT on November
2, between $\sim$09 and 11 UT and $\sim$1430 to 15 UT on November 3, as well
as intermittently throughout both days. The fifth and sixth panels show the
pitch angle spectra for electrons with center energy of 314 and 204 eV,
providing evidence for the ability of these narrow band whistlers to scatter
electrons in this energy range. During the intervals with whistlers, seen in
the electric and magnetic field spectra, there is very significant broadening
of the pitch angle distributions of electrons centered around 314 eV and 204
eV. This feature is most clearly seen around 1230 and 1800 on 2-11-2020, and
$\sim$930 to 1030 and $\sim$13 to $\sim$15 on November 3. Note that some
changes in the pitch angle distributions are associated with changes in the
magnetic field orientation. A detailed discussion of the scattering and
specifics of the resonant mechanisms are presented in Cattell et al. (2020b).
The fourth panel plots the radial component of the proton plasma velocity in
blue (with 300 km/s subtracted to make changes clearer) and the radial
component of the magnetic field in red. The third panel plots magnetic field
in RTN coordinates. As described in Malaspina et al. (2020), the high
frequency electrostatic waves occur primarily in quiet radial magnetic field.
The narrowband whistlers occur primarily within regions with more variable
magnetic field and slightly increased flow, and, at times, within or on the
edges of structures called ‘magnetic switchbacks’ or ‘jets’ (Bale et al.,
2019; Kasper et al., 2019).

Figure 1: Interval during Encounter 1 with narrow band whistler-mode waves and
higher frequency electrostatic waves. Left panels: DC-coupled BPF electric
field spectrum from 12 to 4000 Hz; DC-coupled BBF magnetic field from 12 to
4000 Hz; magnetic field in RTN coordinates, R component of magnetic field in
red with radial component of ion flow -300 km/s in blue. Pitch angle spectra
for electrons with center energy of 314 and 204 eV. Units for the wave spectra
are volts and nT, and for the electron data are eV/cm2s. Right panels:
Spacecraft x component of the electric field (in mV/m) snapshots from seven
different waveform captures during this interval at approximate times
indicated by arrows with blue arrows indicating more than one snapshot. Note
that the time durations vary. See text for details.

One component of the electric field waveforms for seven of the waveform
captures containing narrowband whistlers is plotted in the right hand set of
panels; #1 plots 1 s of a waveform captured at 12:29:20 UT on November 2,
showing an interval of high frequency Bernstein waves followed by whistlers.
The rest of the waveforms were observed on November 3: #2 shows 0.5 s of a
whistler waveform at 10:19:15 UT; #3 plots 0.8 s of a whistler at 10:34:18 UT;
#4 plots the entire 3.5 s capture at 10:43:47 UT to show the packet
modulations, and #5 shows the zoomed in 0.2 s waveform centered on the maximum
amplitude; #6, #7 and #8 plot .2s intervals at 13:49:45 UT, when higher
frequency waves were superimposed on a whistler, at 14:20:26 UT, and at
14:26:11 UT. These examples show the narrowband coherent nature of the
whistler waveforms, as well as the usual duration of individual sub-packets,
and the amplitude modulation. In the statistics presented below, an event is
defined as a 3.5 s wave capture that contains at least one whistler wave
packet. As these examples show (particularly #3 and #4), an event frequently
contains more than one wave packet. Examination of the magnetic field
hodograms (not shown) indicates that the waves are right-hand polarized, as
expected for whistler-mode waves.

The total number of waveform captures containing narrowband whistlers versus
radial distance is plotted in Figure 2, color coded by encounter number. Note
that instrument modes and solar wind conditions varied between encounters, as
did the on-board program for triggering waveform captures. For the 17 waveform
captures with whistlers identified in Encounter 1, when three components of
the search coil data were obtained, the wave vector direction with respect to
the background magnetic field and the solar wind velocity was determined using
minimum variance analysis. The average wave angle with respect to the magnetic
field was 13 degrees, with a maximum of 47 degrees. The average angle is much
less than seen in the STEREO data at 1 AU (Cattell et al., 2020a); note that
there was a very small number of Encounter 1 events compared to the STEREO
database for wave angle determination. Comparison of the electric to magnetic
field ratio for events seen in the bandpass filter data set suggests that
there may be significant number of highly oblique waves. The wave propagation
was very oblique to the solar wind velocity. For 13 sunward-propagating
events, the average angle to the solar wind velocity was 136 degrees; for the
4 anti-sunward cases the angle was 73 degrees. Phase velocities, ranging from
650 km/s to 1550 km/s, were much larger than the solar wind speed.

Figure 2: Number of narrowband whistler wave captures color coded by encounter
number. The number is not normalized by total number of waveform captures
obtained.  Table 1: Average, minimum and maximum values of parameters for
Encounters 1 through 4. Average value is the upper values; minimum and maximum
are given below in parentheses.

Statistics of the properties for the waves identified in the first 4
encounters are shown in Figures 3 and 4. The number of events is not
normalized by total number of waveform captures obtained. Because events in
the statistics are color coded by encounter, the range of parameters observed
in each encounter is shown in Table 1. The average, minimum and maximum values
of the density (cm -3), core electron temperature (eV), suprathermal
temperature (eV), temperature anisotropy, heat flux (Watts/m2),
$\beta_{e\parallel}$, background magnetic field (nT), solar wind speed (km/s)
and distance from the Sun (in solar radii) are shown. There are significant
differences in both the average and extreme values between the different
encounters, as is also clear in the figures below. Events in Encounters 1 and
4 were on average obtained closer to the Sun in regions of higher density and
magnetic field. Events in Encounter 4 were associated with significantly lower
solar wind speeds and higher $\beta_{e\parallel}$. Encounter 2 events were
associated with the largest core temperature, lowest heat flux and lowest
$\beta_{e\parallel}$. Possible explanations for the differences between
Encounters 1 and 2 and Encounter 4 were discussed by Halekas et al. (2020b).

Figure 3 plots the spacecraft frame frequencies at peak power, color coded by
encounter, and the magnitude of the background solar wind magnetic field for
each event. The top panels plot the number of events versus wave frequency, f,
and the number of events versus frequency normalized by electron cyclotron
frequency (f/$f_{ce}$), and the background magnetic field. The bottom panels
plot f, f/$f_{ce}$ and the magnitude of the solar wind magnetic field versus
radial distance from the Sun. There is not a clear radial dependence of the
wave frequency in the spacecraft frame. In contrast to the case at 1 AU, where
Breneman et al. (2010) showed that Doppler shifts were insignificant, there
are sometimes significant Doppler shifts in the waves observed by PSP. For the
Encounter 1 events, for which the shifts could be determined, the shifts
increased the average f/$f_{ce}$ to $\sim$0.2, comparable to the value seen at
1 AU (Cattell et al., 2020a). The lower frequency whistlers described by
Agapitov et al. (2020), utilizing the lower sample rate fields data set, had
significantly larger relative Doppler shifts. The normalized frequency in the
spacecraft frame, f/$f_{ce}$, has a tendency to increase with distance from
the Sun. Further studies including waveform data from other encounters will be
required to determine the effect of Doppler shifts on the radial dependence.
Whistler events usually occurred in regions with reduced magnetic field
magnitudes. The wave amplitudes, determined from the peak amplitude seen in
any component in each event, are plotted in Figure 4, color coded by
encounter. The top panels show the number of whistler captures versus
amplitude of the wave electric field, amplitude of the wave magnetic field and
of the wave magnetic field normalized by background magnetic field
$\left(\delta B_{w}/B_{0}\right)$ where $\delta B_{w}$ is the magnitude of the
wave magnetic field. The bottom panels show the radial dependence of these
amplitudes. There is a clear decrease in wave amplitudes with radial distance
from the Sun, although the decrease in$\left(\delta B_{w}/B_{0}\right)$ is not
as strong. The largest amplitudes were observed close to the Sun during
Encounter 4. Although PSP sees a decrease in the electric field amplitudes
with radial distance, the average amplitude at radial distances around 0.3 AU
is only slightly larger than those observed at 1 AU by STEREO. As noted in
Section 2, the amplitudes for the first three encounters are on average lower
than for Encounter 4, because many waveform captures were triggered by dust
until the algorithm was modified. For this reason, many of the intervals with
whistlers occurred in dust-triggered events rather than ones triggered by wave
amplitude. Data from additional encounters will be required to determine if
the observed amplitude differences between PSP and STEREO are due to
differences in the waveform capture selection criteria or to physics
associated with wave growth and saturation. Note that STEREO did not have a
search coil magnetometer so wave magnetic fields were not directly measured.

Figure 3: Spacecraft frame frequencies for narrowband whistler wave captures
color coded by encounter. Top: Number of events versus frequency, frequency
normalized by electron cyclotron frequency and magnitude of the background
magnetic field. Bottom: Whistler event frequency, frequency normalized by
electron cyclotron frequency and background magnetic field versus radial
distance from the Sun.  Figure 4: Whistler peak amplitudes color coded by
encounter. Left panels: Number of whistler captures versus amplitude of
electric field, magnetic field and magnetic field normalized by background
magnetic field. Right panels: Event amplitude of electric field, magnetic
field and magnetic field normalized by background magnetic field versus radial
distance from the Sun.  Figure 5: Whistler dependence on core density, core
and suprathermal temperature (from the QTN measurement).Top panels plot the
number of events versus core electron density, core and suprathermal
temperature. Bottom panels plot the same quantities versus radial distance
from the Sun.

The association of the whistler events with electron parameters is shown in
Figure 5. For most waveform captures, the electron parameters were determined
within a few seconds of the capture, with median times of $\sim$ 2 s for QTN
parameters and $\sim$ 4 s for SWEAP-determined parameters. For almost all
events, the ratio of the electron cyclotron frequency to the electron plasma
frequency (not shown)is $<$0.01. The left hand panels plot the number of
events versus core electron density, core and suprathermal electron
temperature, and the right panels plot these quantities versus radial distance
from the Sun. These comparisons, which are restricted to events with
narrowband whistler waves, exhibit interesting differences from the results
for all intervals during the encounters 1 and 2 presented by Moncuquet et al.
(2020). Their results show that the core electron temperature decreases with
radial distance, and the suprathermal temperature was almost constant. For
intervals with the waves, we see a slight increase in the core temperature,
possibly indicating heating of core electrons by the waves, and a slight
decrease in the suprathermal temperature. Note that our statistics are small
and the observed variability at a given radial distance is as large as the
average change with radial distance.

Possible instability mechanisms are examined in Figures 6. The left panel of
Figure 6 shows temperature anisotropy versus parallel electron beta. The upper
red line is the whistler temperature anisotropy
threshold,$\frac{T_{e\bot}}{T_{e\parallel}}$ = 1+0.27/$\beta_{e\parallel}$0.57
and the lower red line is an arbitrary firehose instability (both from Lacombe
et al. (2014), based on Gary et al. (1999)). The middle panel plots the
normalized electron heat flux versus $\beta_{e\parallel}$ with the linear
instability threshold for the heat flux fan instability (Vasko et al., 2019,
for the parameters of 0.5 and 1 in their Table 1). The most striking feature
is that the waves occur when $\beta_{e\parallel}$¿1. Many of the largest
amplitude waves occurred in Encounter 4, which had significantly higher
$\beta_{e\parallel}$. Halekas et al. (2020a) showed that during encounters 1
and 2 inside 0.24 AU, $\beta_{e\parallel}$ was usually ¡1. This association of
narrowband whistler waves with $\beta_{e\parallel}$ $>$1 was also seen in the
STEREO data at 1 AU. The wave occurrence is constrained by both the whistler
temperature anisotropy threshold and the heat flux fan instability threshold,
as was also the case at 1 AU (Cattell et al., 2020a). The temperature
anisotropy destabilizes parallel propagating waves at lower frequencies that
we observe. We conclude, therefore, that the waves are most likely driven by
the fan instability. This is consistent with the study of the electron heat
flux and beta for all intervals inside 0.25 AU in the first five encounters by
Halekas et al. (2020b), and with the conclusion of Agapitov et al. (2020) for
a set of events in Encounter 1. Note, however, that there are a significant
number of cases with large $\beta_{e\parallel}$ where the normalized heat flux
is above the threshold (most from Encounter 4).

Figure 6:  Comparison to instability mechanisms. From left to right: The
temperature anisotropy versus parallel electron beta for wave events. The
upper red line is the whistler temperature anisotropy
threshold,$\frac{T_{e\bot}}{T_{e\parallel}}$ = 1+0.27/$\beta_{e\parallel}$0.57
and the lower red line is an arbitrary firehose instability (both from Lacombe
et al. 2014, based on Gary et al., 1999). Normalized electron heat flux vs
$\beta_{e\parallel}$. The yellow and red lines plot the linear instability
thresholds for the whistler heat flux fan instability from equation 5 of Vasko
et al.(2019) for the parameters of .5 and 1 in their Table 1). The electron
Alfven energy to compare to the threshold for the beam mechanism of Sauer and
Sydora (2010)

.

The third panel of Figure 6 plots the energy associated with the electron
Alfven speed for the whistler events for comparison to the electron beam
driven instability proposed by (Sauer & Sydora, 2010). This mechanism which
generates highly oblique waves requires electron beams that propagate at
speeds greater than twice the electron Alfven speed. The energy associated
with the Alfven speed is very low for the whistler events, $\sim$10 to 20 eV;
thus this mechanism would require beams with energies of $\sim$40 to 80 eV.
This is an order of magnitude lower than would be required for the mechanism
to operate at 1 AU. To date, we have not yet been able to identify beam
features at the appropriate energies in either event list.

To better assess the occurrence probability of these waves we utilized one
electric field channel in the DC coupled spectral data, at 30 s resolution. We
examined by eye the spectral data for each hour during the first encounter
interval shown in the BPF data in Figure 1, which covers 31 hours on November
2 and 3. This yields only a very rough estimate of occurrence rate. The
individual waveform captures (duration 3.5 s) usually contain several
individual wave packets. An example of a 3.5 s waveform capture was shown
above in Figure 1, packet #4. The large amplitude whistler packets have
durations the order of a few seconds (see #4, and the shorter duration
waveforms in Figure 1), while the lower amplitude waves can last through an
entire capture. Waveform #1 in Figure 1 provides an example when the higher
frequency Bernstein waves had the largest amplitude for the initial $\sim$1 s,
followed by an interval with both wave types with whistler dominating from
$\sim$1 s to 2 s. We have also have examined the higher cadence BPF data,
which can more accurately determine the duration of regions with whistler
packets. The BP cadence is comparable to the duration of the observed large
amplitude whistler packets. Comparison of the spectral (1 sample /64 Cy) to
the BPF data (1 sample/2 Cy) suggests that the occurrence could be on the
order of five to 10 percent, based on the electric field spectra.There are a
significant number of waves that are observed only in the electric field,
consistent with very oblique propagation. Any definitive determination of an
occurrence rate will depend on the amplitude threshold selected.

##  4 Discussion and conclusions

We have compared statistics of the properties of the narrowband whistler-mode
waves observed in waveform capture data from Parker Solar Probe during the
first four encounters inside $\sim$0.3 AU, to properties observed in waveform
capture data from STEREO at 1 AU. At both radial distances, the waves are
narrowband and large amplitude. The association with heat flux and beta is
generally consistent with the whistler fan instability. In both data sets the
whistlers are observed only for beta ¿1, and the average temperature
anisotropy was $\sim$.9. The PSP electron data show significant scattering at
strahl energies, as documented in detail by Cattell et al. (2020b). This is
consistent with a study of electron heat flux (Halekas et al., 2020b) for
Encounters 1 through 5, which showed that the heat flux and beta were
constrained by the fan instability threshold, providing evidence that these
waves regulate the electron heat flux.

Many instability mechanisms have been proposed for whistler-mode waves in the
solar wind that have free energy sources associated with electron properties,
including electron temperature anisotropies (Gary & Wang, 1996), heat flux
(Feldman et al., 1975; Gary et al., 1994) , heat flux fan instability (Krafft
& Volokitin, 2010; Vasko et al., 2019), the fast magnetosonic/whistler mode
(Verscharen et al., 2019), and electron beam instability (Sauer & Sydora,
2010). Theoretical studies of the dispersion relations have concluded that the
most unstable modes for the temperature anisotropy and heat flux instabilities
are parallel propagating, and have frequencies f/$f_{ce}$ of $\sim$.01, lower
than the frequencies we observe. The transition to oblique modes (Gary et al.,
2011) occurs at values of beta that are much smaller than those observed in
the PSP data presented herein. Only the heat flux fan instability, the
magnetosonic/whistler mode, and the beam instability have the highest growth
rates at oblique angles. The PSP whistlers occurred when beta was high, and
the events for which Doppler shifts could be determined, had frequencies of
f/$f_{ce}$ of $\sim$ 0.2. Most cases, however, propagated within 20 degrees of
the magnetic field, and none were propagating close to the resonance cone as
was seen by Agapitov et al. (2020) in a study of lower frequency whistlers in
Encounter 1 and byCattell et al. (2020a) at 1 AU. The magnetosonic/whistler
mode is low beta, and the most unstable modes are at higher frequencies
(f/$f_{ce}$ $\sim$.5) than observed at PSP (Verscharen et al., 2019). No beam
features have been observed in energy range needed for the whistler beam
instability. The whistler fan instability is most unstable in the range of
f/$f_{ce}$ of $\sim$ 0.1 to 0.2 (Vasko et al., 2019), in the range we observe.
As shown in Figure 7, the wave occurrence was constrained by the heat fan flux
instability threshold. For these reasons, we conclude that the whistler-mode
waves observed by PSP are most likely due to the fan instability, as was also
the case for the STEREO whistlers (Cattell et al., 2020a). Although the wave
occurrence is also constrained by whistler temperature anisotropy, the
observed wave properties are not usually consistent with this mode. As
discussed below, however, it may be the case that the parallel whistlers and
the very oblique whistlers are associated with different instability or
saturation mechanisms.

At 1 AU, two distinct populations of whistler-mode waves with frequencies of
f/$f_{ce}$ of $\sim$0.1 to .2 have been reported from waveform capture data;
one population is parallel-propagating with small electric field amplitudes
(Lacombe et al., 2014; Graham et al., 2017; Tong et al., 2019) and one is
obliquely propagating with resultant large electric fields (Breneman et al.,
2010; Cattell et al., 2020a). Although the parallel-propagating waves are
usually seen in quiet, slow solar wind (Lacombe et al., 2014), Tong et al.
(2019) show that quasi-parallel whistlers can also occur in the faster solar
wind. The oblique waves are often seen in faster solar wind (Breneman et al.,
2010; Cattell et al., 2020a)(see Figure 7). The Parker Solar Probe data shown
herein included both parallel and oblique waves, as was also described in
(Agapitov et al., 2020). There is a tendency for more electrostatic whistlers
(i.e. more oblique)to occur within regions of enhanced flow. A recent study of
frequency bank spectral data from Helios (Jagarlamudi et al., 2020) presented
statistics on waves with spacecraft frame frequencies between $\sim f_{lh}$
and 0.5 $f_{ce}$, identified in search coil magnetic field at distances of 0.3
to 0.9 AU. The observed spectral peaks were identified as whistler-mode based
on similarities to Lacombe et al. (2014), but polarization, wave vectors and
Doppler shifts could not be determined. The waves were observed almost
exclusively in the slow solar wind ($<$400 km/s).

Figure 7 plots histograms of the number of whistler events versus solar wind
speed in the Cattell et al. (2020a) STEREO database (left) and the number of
PSP events color coded by encounter (right) versus solar wind speed. The
center panel plots the PSP events versus solar wind speed and radial distance.
The highly oblique whistlers observed at 1 AU by STEREO are predominantly seen
with solar wind speeds of $\sim$400 km/s, but are also observed with speeds up
to $\sim$700 km/s. PSP events are associated with lower solar wind speeds
($\sim$300 km/s). The bi-modal distribution is likely due to to the small
number of events, and to radial distance effects, differences in conditions
during each encounter, as indicated by the center panel which plots the PSP
events versus solar wind speed and radial distance. Encounter 4 events were
all obtained inside $\sim$50 solar radii and solar wind speeds were $\sim$200
km/s, whereas events during Encounters 2 and 3 were primarily outside $\sim$50
solar radii with solar wind speeds of $\sim$350 km/s(see also Table 1). The
contrast in solar wind conditions and heat flux during Encounter 4 compared to
Encounters 1 and 2 was described by Halekas et al. (2020b).  The differences
in wave association with solar wind speed between PSP events inside .3 AU and
the STEREO events at 1 AU may just be due to the evolution of the solar wind.
The Jagarlamudi et al. (2020) observations which cover the distances between
.3 AU and .9 AU, however, were associated with slow flow. Wave vector angles
have been determined for only a small fraction of the PSP events; therefore,
it is not yet possible to determine if there is a relationship between wave
obliquity and solar wind speed at these radial distances. The parallel
propagating waves and the oblique waves may represent two different modes, or
different sources of free energy. However, the distinction may also be due
differences in instrumentation. Future studies utilizing the Parker Solar
Probe data set may resolve the relationship between the parallel and highly
oblique waves.

Figure 7: The relationship of whistler occurrence to solar wind speed on PSP
and STEREO. From left to right: the number of events versus solar wind speed
on STEREO from the Cattell et al. (2020a) database, PSP whistler events versus
solar wind velocity and radial distance, and number of PSP whistler events
versus solar wind speed.

There are two main differences between the characteristics of the whistlers
identified in waveform captures inside 0.2 AU and the waves at 1 AU: (1) the
association with larger scale solar wind properties and structure; and (2) the
occurrence of a broader band less coherent mode at 1 AU that has not yet been
identified in PSP waveform data. In addition, inside 0.2 AU the narrowband
electromagnetic whistlers are interspersed with lower amplitude electrostatic
whistler-mode waves and Bernstein waves at frequencies of $\sim$0.7 $f_{ce}$
to $>$$f_{ce}$ (Malaspina et al., 2020; Bale et al., 2019), which have not
been observed at 1 AU in the STEREO waveform data.

At 1 AU, the narrowband oblique whistlers are most often associated with SIRs,
often filling the downstream region of increased solar wind speed, and often
variable magnetic field. The waves are also seen within CMEs (Cattell et al.,
2020a). As shown in Figure 1, inside $\sim$0.3 AU, the whistlers are
associated with intervals of variable background magnetic field and slight
increases in solar wind flow, sometimes due to magnetic field switchbacks. The
association with switchbacks was previously described by Agapitov et al.
(2020). The intervals with packets of narrowband whistlers can last for
several hours, but not for a day or more as seen at 1 AU. This is most likely
due to the much more variable solar wind conditions observed by PSP close to
the Sun. Future studies including data from additional encounters will examine
whether there is an association of the whistlers with SIRs or CMEs inside
$\sim$.3 AU.

Both the wave magnetic field amplitudes normalized to the background magnetic
field and the electric field amplitudes observed by PSP decrease with radial
distance from the Sun, however, the average electric field amplitudes observed
by STEREO at 1 AU are comparable to those seen by PSP near .3 AU. This may be
due to the different selection criteria for burst data for the two spacecraft
or to differences in the physics. For example, the waves may, on average, be
more oblique at 1 AU or the wave growth and saturation mechanisms may be
different due to differences in the solar wind and electron properties.

Initial results on whistler-mode waves observed by Parker Solar Probe
identified in the magnetic field data during the first encounter were
presented by Agapitov et al. (2020), utilizing the $\sim$300 samples/s
waveform data. The observed waves had large amplitudes ($\sim$2 to 4 nT),
often propagated towards the sun, were significantly Doppler shifted with
plasma frame frequencies of $\sim$0.2 to 0.5 $f_{ce}$, and variable wave
angles. The differences including average f/$f_{ce}$ warrant additional
studies for other encounters. This will require developing a method to
accurately determine wave vector direction when only two components of the
search coil data are available, utilizing an approach similar to that used on
STEREO based on the three components of the electric field waveform and the
cold plasma dispersion relation (Cattell et al., 2008).

In conclusion, we have shown that narrowband whistler mode waves observed in
the PSP waveform capture data inside $\sim$.3 AU have many characteristics
similar to those seen by STEREO at 1 AU. In both regions the waves are most
consistent with the whistler heat flux fan instability and occur when beta is
greater than 1. The waves at 1 AU have slightly higher average f/$f_{ce}$ and
are on average more oblique, but both of these differences may be due to the
small number of PSP events for which we have determined wave angle and Doppler
shifts. When there are wave events, the radial dependence of the core and
suprathermal temperatures is different from that seen for the full electron
data set (Halekas et al., 2020a), possibly indicating that the waves heat core
electrons. At PSP, the waves are often associated with variable magnetic field
and slightly enhanced solar wind flow, sometimes with ‘switchbacks,’ whereas
at 1 AU, the waves are most often seen in the downstream region of SIRs, also
regions of enhanced flow. Inside $\sim$.3 AU, the regions containing wave
packets tend to last for intervals of hours, whereas at 1 AU, they can last
for days. It is very likely these differences are due to the fact that the
solar wind is much more variable on short time scales at PSP compared to at 1
AU.The waves are associated with scattering of strahl energy electrons. A very
rough estimate of the wave occurrence at PSP suggests that the waves are often
the dominant wave mode at frequencies below $\sim$3 kHz; combined with the
observations of scattering, this suggests that the narrowband whistlers may
play a significant role in the evolution of solar wind electrons and
regulation of heat flux.

###### Acknowledgements.

We acknowledge the NASA Parker Solar Probe Mission, and the FIELDS team led by
S. D. Bale, and the SWEAP team led by J. Kasper for use of data. The FIELDS
experiment on the Parker Solar Probe spacecraft was designed and developed
under NASA contract NNN06AA01C. Work at University of Minnesota and at
University of Iowa was supported under the same contract.

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# A generalization of the Shafer-Fink inequality

Jacopo D’Aurizio
Università di Pisa
<EMAIL_ADDRESS>
(02-03-2013)

In this article we will prove some generalizations and extensions of the
Shafer-Fink ([3]) double inequality for the arctangent function:

######  Theorem 1.

For any positive real number $x$,

$\frac{3x}{1+2\sqrt{1+x^{2}}}<\arctan x<\frac{\pi x}{1+2\sqrt{1+x^{2}}}$

holds.

###### Proof.

Following the lines of ([2]), we consider the substitution $x=\tan\theta$,
that gives the following, equivalent form of the inequality:

$\forall\theta\in
I=(0,\pi/2),\qquad\theta(\cos\theta+2)-\pi\sin\theta<0<\theta(\cos\theta+2)-3\sin\theta.$

If now we set

$f_{K}(\theta)=(\cos\theta+2)-K\,\frac{\sin\theta}{\theta}$

we have:

$\theta^{2}\,\frac{df_{K}}{d\theta}=(K-\theta^{2})\sin\theta-K\theta\cos\theta.$

Since for any $\theta\in I$ we have:

$\frac{\theta}{\tan\theta}<1-\frac{\theta^{2}}{3}<1-\frac{\theta^{2}}{\pi},$

$f_{3}(\theta)$ ed $f_{\pi}(\theta)$ are both non-decreasing on $I$, in virtue
of $\frac{df_{K}}{d\theta}\geq 0$; moreover, $f_{K}^{\prime}(0)=0$ and
$f_{K}^{\prime}$ cannot be zero on $I$. Since:

$f_{3}(0)=0,\quad f_{3}(\pi/2)>0,\quad f_{\pi}(0)<0,\quad f_{\pi}(\pi/2)=0,$

the claim follows. ∎

We give now a different proof of this inequality, that relies on the bisection
formula for the cotangent function and the associated Weierstrass product.

From the logarithmic derivative of the Weierstrass product for the sine
function we know that for any $x\in[0,\pi/2]$

$f(x)=x\cot x=1-2\sum_{k=1}^{+\infty}\frac{\zeta(2k)}{\pi^{2k}}\,x^{2k}$

holds. Since $f(x)$ is an even function, there exists a suitable linear
combination $g_{1}(x)$ of $f(x)$ and $f(x/2)$ that satisfies:

$g_{1}(x)=A_{0}f(x)+A_{1}f(x/2)=1-\sum_{k\geq 2}C^{(1)}_{k}\,x^{2k}.$

With the choices $A_{0}=-\frac{1}{3},A_{1}=\frac{4}{3}$ the previous identity
holds, and, for any $k\geq 2$:

$C^{(1)}_{k}=\left(A_{0}+\frac{A_{1}}{4^{k}}\right)\frac{\zeta(2k)}{\pi^{2k}}<0,$

so $g_{1}(x)$ is an increasing and convex function over $I=[0,\pi/2]$. From
that,

$\forall x\in I,\quad\left(-\frac{1}{3}x\cot
x+\frac{2}{3}x\cot\frac{x}{2}\right)\in[g_{1}(0),g_{1}(\pi/2)]=[1,\pi/3]$

follows. If now we consider the bisection formula for the cotangent function:

$\cot\frac{x}{2}=\cot x+\sqrt{1+\cot^{2}x}$

we have a different proof of the Shafer-Fink inequality.

We consider now $g_{2}(x)$ as a linear combination of $f(x),f(x/2)$ and
$f(x/4)$ such that:

$g_{2}(x)=A_{0}f(x)+A_{1}f(x/2)+A_{2}f(x/4)=1-\sum_{k\geq
3}C^{(2)}_{k}\,x^{2k}.$

From the annihilation of the coefficient of $x^{2}$ in the RHS we deduce the
constraint $A_{0}+A_{1}\cdot\frac{1}{4}+A_{2}\cdot\frac{1}{16}=0$, and from
the annihilation of the coefficient of $x^{4}$ we deduce the constraint
$A_{0}+A_{1}\cdot\frac{1}{16}+A_{2}\cdot\frac{1}{256}=0$. If we take
$p_{2}(x)=A_{0}+A_{1}x+A_{2}x^{2}$, such constraints translate into
$p_{2}(1/4)=p_{2}(1/16)=0$, from which:

$p_{2}(x)=K_{2}\left(x-\frac{1}{4}\right)\left(x-\frac{1}{16}\right),$

with $K_{2}=(1-1/4)^{-1}\cdot(1-1/16)^{-1}$ in order to grant
$A_{0}+A_{1}+A_{2}=p_{2}(1)=1$.
Since $C^{(2)}_{k}=\frac{\zeta(2k)}{\pi^{2k}}p_{2}(4^{-k})$, all the non-zero
coefficients of the Taylor series of $g_{2}(x)$, except (at most) the first
one, have the same sign, so $g_{2}(x)$ is a monotonic function over $I$. In
particular:

$\displaystyle\forall x\in
I,\qquad\frac{\pi(3+8\sqrt{2})}{45}=g_{2}(\pi/2)\leq g_{2}(x)$
$\displaystyle=\frac{1}{45}\left(f(x)-20f(x/2)+64f(x/4)\right)$
$\displaystyle=\frac{x}{45}\left(\cot x-10\cot(x/2)+16\cot(x/4)\right)\leq 1,$

from which we get:

$\pi(3+8\sqrt{2})\leq x\left(\cot x-10\cot(x/2)+16\cot(x/4)\right)\leq 45.$

By using twice the bisection formula for the cotangent, we have the following
strengthening of the Shafer-Fink inequality:

######  Theorem 2 (D’Aurizio).

For any positive real number $x$

$\pi(3+8\sqrt{2})\cdot f(x)<\arctan x<45\cdot f(x)$

holds, where:

$f(x)=\frac{x}{7+6\,\sqrt{1+x^{2}}+16\sqrt{2}\,\sqrt{1+x^{2}+\sqrt{1+x^{2}}}}.$

The same approach leads to an arbitrary strengthening of the Shafer-Fink
inequality:

######  Theorem 3 (D’Aurizio).

For any positive real number $x$ and for any positive natural number $n$,
once defined:

$f(x)=x\cot x=1-2\sum_{k=1}^{+\infty}\frac{\zeta(2k)}{\pi^{2k}}\,x^{2k},$
$p_{n}(x)=\prod_{k=1}^{n}\frac{(4^{k}x-1)}{(4^{k}-1)}=A_{0}+A_{1}x+\ldots+A_{n}x^{n},$
$g_{n}(x)=\sum_{k=0}^{n}A_{k}\,f(2^{-k}x)=x\sum_{k=0}^{n}\frac{A_{k}}{2^{k}}\,\cot(2^{-k}x),$
$e_{j}(x_{1},\ldots,x_{k})=\sum_{sym}x_{1}\cdot\ldots\cdot x_{j},$
$L_{0}(x)=1,\qquad L_{n+1}(x)=L_{n}(x)+\sqrt{x^{2}+L_{n}(x)^{2}},$

we have:

$K_{low}\cdot a_{n}(x)<\arctan(x)<K_{high}\cdot a_{n}(x),$

where $K_{low}=\min(g_{n}(0),g_{n}(\pi/2))$,
$K_{high}=\max(g_{n}(0),g_{n}(\pi/2))$ and:

$a_{n}(x)=x\cdot\left(\sum_{j=0}^{n}(-1)^{n-j}\cdot L_{j}(x)\cdot 2^{j}\cdot
e_{j}(1,4,\ldots,4^{n-1})\right)^{-1}.$

Moreover, $K_{high}-K_{low}<\frac{1}{4^{n}}$.

###### Proof.

By taking

$p_{n}(x)=\prod_{k=1}^{n}\frac{(4^{k}x-1)}{(4^{k}-1)}=A_{0}+A_{1}x+\ldots+A_{n}x^{n}$

we have $p_{n}(1)=1$ and $p_{n}(4^{-j})=0$ for every $j\in[1,n]$. In
particular, the Taylor series of

$g_{n}(x)=\sum_{k=0}^{n}A_{k}\,f(2^{-k}x)=x\sum_{k=0}^{n}\frac{A_{k}}{2^{k}}\,\cot(2^{-k}x).$

is equal to:

$1-2\sum_{k=1}^{+\infty}\frac{\zeta(2k)p_{n}(4^{-k})}{\pi^{2k}}\,x^{2k}=1-2\sum_{k>n}C^{(k)}_{n}\,x^{2k},$

and all the $C^{(k)}_{n}$ with $k>n$ have the same sign, so $g_{n}(x)$ is
monotonic over $[0,\pi/2]$, with $g_{n}(0)=1$.
In particular, we have:

$\forall x\in[0,\pi/2],\qquad
x\cdot\sum_{j=0}^{n}(-1)^{n-j}\cot\left(\frac{x}{2^{j}}\right)2^{j}\,e_{j}(1,4,\ldots,4^{n-1})\leq\prod_{k=1}^{n}(4^{k}-1),$

where $e_{j}$ is the $j$-th elementary symmetric function. Since for any $m>n$
we have $|p_{n}(4^{-m})|<1$,

$\left|g_{n}(\pi/2)-g_{n}(0)\right|\leq\sum_{k>n}\frac{\zeta(2k)}{4^{k}}<\frac{1}{4^{n}}.$

holds. ∎

We give now another upper bound for the arctangent function that does not
belong to the last family of inequalities, but that strenghtens the inequality
$\arctan x<\frac{\pi x}{1+2\sqrt{1+x^{2}}}$, too.

######  Theorem 4.

For any positive real number $x$

$\arctan x<\frac{\pi x}{\frac{4}{\pi}+\sqrt{2}\sqrt{1+x^{2}+x\sqrt{1+x^{2}}}}$

holds.

###### Proof.

By using the substitution $x=\tan\theta$, it is sufficient to prove that for
any $\theta\in I=[0,\pi/2]$ we have:

$\theta\leq\frac{\pi\sin\theta}{\frac{4}{\pi}\cos\theta+\sqrt{2+2\sin\theta}},$

that is also equivalent, up to the change of variable $\theta=\pi/2-\phi$, to
the inequality:

$\frac{\pi}{2}-\phi\leq\frac{\pi\cos\phi}{\frac{4}{\pi}\sin\phi+2\cos(\phi/2)},$

or the inequality:

$\frac{\cos\phi}{1-\frac{2\phi}{\pi}}\geq\cos(\phi/2)\left(\frac{4}{\pi}\sin(\phi/2)+1\right).$

In order to prove the latter it is sufficient to prove:

$\frac{\cos\phi}{1-\frac{2\phi}{\pi}}\geq\cos(\phi/2)\left(1+\frac{2\phi}{\pi}\right),$

or:

$\frac{\cos\phi}{1-\frac{4\phi^{2}}{\pi^{2}}}\geq\cos(\phi/2).$

By considering the Weierstrass product for the cosine function we may rewrite
the last line in the form:

$\prod_{k=1}^{+\infty}\left(1-\frac{4x^{2}}{(2k+1)^{2}\pi^{2}}\right)\geq\prod_{k=1}^{+\infty}\left(1-\frac{x^{2}}{(2k-1)^{2}\pi^{2}}\right).$

By considering the Taylor series of the logarithm of both sides, we simply
have to prove:

$\forall
m\in\mathbb{N}_{0},\qquad(4^{m}-1)\zeta(2m)-4^{m}-(1-4^{-m})\zeta(2m)\leq 0,$

that is a consequence of:

$\forall m\in\mathbb{N}_{0},\qquad\zeta(2m)\leq\frac{4^{m}+1}{4^{m}-1},$

implied by:

$\forall m\in\mathbb{N}_{0},\qquad(4^{m}-1)(\zeta(2m)-1)\leq 2.$

An upper bound for the LHS is the series:

$1+\sum_{k=1}^{+\infty}\left(\frac{4}{(2k+1)^{2}}\right)^{m},$

whose value decreases as $m$ increases; so we have:

$(4^{m}-1)(\zeta(2m)-1)\leq
1+\sum_{k=1}^{+\infty}\frac{4}{(2k+1)^{2}}=3\zeta(2)-3,$

and the RHS is less than $2$ since $\pi^{2}<10$ holds. ∎

Now we make a step back into the general setting of double inequalities for
the arctangent function.

######  Lemma 1.

If $f(u),g(u)$ are a couple of real functions such that, for any $u\in[0,1]$,

$f(u)\leq\arctan u\leq g(u)$

holds, then:

$2\cdot f\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\leq\arctan x\leq 2\cdot
g\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)$

holds for any $x\in\mathbb{R}^{+}$.

###### Proof.

In virtue of the angle bisector theorem,

$\arctan t=2\arctan\left(\frac{t}{1+\sqrt{1+t^{2}}}\right)$

for any $t\geq 0$, so if the first inequality holds for any $\theta=\arctan u$
in the range $[0,\pi/4]$, the second inequality holds for any $\theta=\arctan
x$ in the range $[0,\pi/2]$. ∎

The last lemma gives a third way to prove the Shafer-Fink inequality. By
direct inspection of the Taylor series of $\frac{\arctan u}{u}$, it is easy to
show that $(3+u^{2})\frac{\arctan u}{u}$ is an increasing function over
$[0,1]$, so:

$\frac{3u}{3+u^{2}}\leq\arctan u\leq\frac{\pi u}{3+u^{2}},$

and it is sufficient to use the substitution $u=\frac{x}{1+\sqrt{1+x^{2}}}$ to
give another proof of the Shafer-Fink inequality.

######  Lemma 2.

If an approximation $f(u)$ of the arctangent function satisfies:

$\|f(u)-\arctan(u)\|_{\mathbb{R}^{+}}=\sup_{u\in\mathbb{R}^{+}}|f(u)-\arctan(u)|=C_{\infty},$

then

$\left\|2\cdot
f\left(\frac{u}{1+\sqrt{1+u^{2}}}\right)-\arctan(u)\right\|_{\mathbb{R}^{+}}=2\cdot\|f(u)-\arctan(u)\|_{(0,1)}=2\cdot
C_{1},$

and, for any $t\in(0,1)$,

$\left\|2\cdot
f\left(\frac{u}{1+\sqrt{1+u^{2}}}\right)-\arctan(u)\right\|_{(0,t)}=2\cdot\|f(u)-\arctan(u)\|_{\left(0,\frac{2t}{1-t^{2}}\right)}.$

This simple consequence of the previous lemma tell us the fact that any
algebraic approximation of the arctangent function in a right neighbourhood of
zero can be “lifted” to an algebraic approximation over the whole
$\mathbb{R}^{+}$, through the iteration of the map

$f(u)\quad\longrightarrow\quad 2\cdot
f\left(\frac{u}{1+\sqrt{1+u^{2}}}\right).$

For example, if we consider the Lagrange interpolation polynomial for the
arctangent function with respect to the points
$(0,\tan(\pi/8)=\sqrt{2}-1,\tan(\pi/4)=1)$

$p(x)=\frac{\pi}{4}\cdot\frac{x(x-\sqrt{2}+1)}{2-\sqrt{2}}+\frac{\pi}{8}\cdot\frac{x(x-1)}{(\sqrt{2}-1)(\sqrt{2}-2)},$

we have

$\|p(x)-\arctan x\|_{(0,1)}<\frac{1}{230},$

so, by considering $2\cdot p\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)$:

######  Theorem 5.

For any non negative real number $x$, the absolute difference between
$\arctan(x)$ and

$\frac{\pi
x\left(\left(4+\sqrt{2}\right)\left(1+\sqrt{1+x^{2}}\right)-\sqrt{2}\,x\right)}{8\left(1+\sqrt{1+x^{2}}\right)^{2}}$

is less than $\frac{1}{115}$.

Another way to produce really effective approximation is to use the Chebyshev
expansion for the arctangent function:

######  Lemma 3.

The sequence of functions:

$f_{n}(x)=2\sum_{k=0}^{n}\frac{(-1)^{k}}{(2k+1)(1+\sqrt{2})^{2k+1}}\;T_{2k+1}(x),$

where $T_{k}(x)$ is the $k$-th Chebyshev polynomial of the first kind, gives a
uniform approximation of the arctangent function over the interval $[0,1]$:

$\|\arctan x-f_{n}(x)\|_{[0,1]}\leq\frac{1}{(1+\sqrt{2})^{2n+3}}.$

Moreover,

$\arctan(mx)=2\sum_{k=0}^{+\infty}\frac{(-1)^{k}}{(2k+1)}\left(\frac{m}{1+\sqrt{1+m^{2}}}\right)^{2k+1}\,T_{2k+1}(x)$

holds for any $x\in(-1,1)$ and for any $m\in\mathbb{N}_{0}$.

######  Theorem 6.

For any $n\in\mathbb{N}_{0}$ and for any $x\in\mathbb{R}$

$\left|\;\arctan
x-4\sum_{k=0}^{n}\frac{(-1)^{k}}{(2k+1)(1+\sqrt{2})^{2k+1}}\;T_{2k+1}\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\right|\leq\frac{1}{\left(3+2\sqrt{2}\right)^{n}}.$

Still another way is to use the continued fraction representation for the
arctangent funtion:

$\arctan
z=\frac{z}{1+\frac{z^{2}}{3+\frac{4z^{2}}{5+\frac{9z^{2}}{7+\frac{16z^{2}}{9+\frac{25z^{2}}{11+\ldots}}}}}},$

from which we get a sequence of approximations for $\arctan x$ over $[0,1]$:

$\left\\{\begin{array}[]{cll}K_{1}(x)&=\displaystyle\frac{x}{1+x^{2}/3},&\\\\[11.38092pt]
K_{2}(x)&=\displaystyle\frac{x}{1+x^{2}/(3+4x^{2}/5)}&=\displaystyle\frac{x(15+4x^{2})}{15+9x^{2}},\\\\[11.38092pt]
K_{3}(x)&=\displaystyle\frac{x}{1+x^{2}/(3+4x^{2}/(5+9x^{2}/7))}&=\displaystyle\frac{5x\left(21+11x^{2}\right)}{105+90x^{2}+9x^{4}}\\\\[11.38092pt]
\ldots&&\end{array}\right.$

that satisfy:

$\|\arctan x-K_{n}(x)\|_{[0,1]}\leq\frac{1}{2\cdot 4^{n}},$

so:

$\left\|\arctan
x-K_{n}\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)\right\|_{\mathbb{R}}\leq\frac{1}{4^{n}},$

with an error term that is the same achieved by $a_{n}(x)$, defined as in
Theorem (3).

By using the Taylor series for the arctangent function with respect to the
point $x=1$ one has:

$\arctan{x}=\frac{\pi}{4}-\sum_{j=0}^{+\infty}\left(-\frac{(1-x)^{4}}{4}\right)^{j}\cdot\left(\frac{(1-x)}{2(4j+1)}+\frac{(1-x)^{2}}{2(4j+2)}+\frac{(1-x)^{3}}{4(4j+3)}\right).$

By plugging in $x=2/3$ we have:

$\arctan\frac{1}{5}=\sum_{j=0}^{+\infty}\left(-\frac{1}{324}\right)^{j}\cdot\left(\frac{1}{6(4j+1)}+\frac{1}{18(4j+2)}+\frac{1}{108(4j+3)}\right),$

and by plugging in $x=119/120$ we have:

$\arctan\frac{1}{239}=\sum_{j=0}^{+\infty}\left(-\frac{1}{829440000}\right)^{j}\cdot\left(\frac{1}{240(4j+1)}+\frac{1}{28800(4j+2)}+\frac{1}{6912000(4j+3)}\right).$

The Machin Formula

$\frac{\pi}{4}=4\arctan\frac{1}{5}+\arctan\frac{1}{239}$

give us the possibility to exhibit a good approximation for $\pi$:

$\displaystyle\pi=8\;$
$\displaystyle\sum_{j=0}^{+\infty}\left(-\frac{1}{324}\right)^{j}\cdot\left(\frac{1}{3(4j+1)}+\frac{1}{9(4j+2)}+\frac{1}{54(4j+3)}\right)+$
$\displaystyle+$
$\displaystyle\sum_{j=0}^{+\infty}\left(-\frac{1}{829440000}\right)^{j}\cdot\left(\frac{1}{60(4j+1)}+\frac{1}{7200(4j+2)}+\frac{1}{1728000(4j+3)}\right).$

In the same fashion, we have that:

$\arctan\frac{1}{2z-1}=\sum_{j=0}^{+\infty}\left(-\frac{1}{4z^{4}}\right)^{j}\cdot\left(\frac{1}{2z(4j+1)}+\frac{1}{2z^{2}(4j+2)}+\frac{1}{4z^{3}(4j+3)}\right)$

holds for any $z\geq 1$, and the truncated series gives a better and better
approximation as $z$ goes to infinity. By a change of variable, the same is
true for:

$\arctan\frac{1}{t}=\sum_{j=0}^{+\infty}\left(-\frac{4}{(t+1)^{4}}\right)^{j}\cdot\left(\frac{1}{(t+1)(4j+1)}+\frac{2}{(t+1)^{2}(4j+2)}+\frac{2}{(t+1)^{3}(4j+3)}\right),$

and:

$\arctan
u=\sum_{j=0}^{+\infty}\left(-\frac{4u^{4}}{(u+1)^{4}}\right)^{j}\cdot\left(\frac{u}{(u+1)(4j+1)}+\frac{2u^{2}}{(u+1)^{2}(4j+2)}+\frac{2u^{3}}{(u+1)^{3}(4j+3)}\right)$

holds for any $u\in[0,1]$. By taking:

$s_{n}(u)=\sum_{j=0}^{n}\left(-\frac{4u^{4}}{(u+1)^{4}}\right)^{j}\cdot\left(\frac{u}{(u+1)(4j+1)}+\frac{2u^{2}}{(u+1)^{2}(4j+2)}+\frac{2u^{3}}{(u+1)^{3}(4j+3)}\right)$

we have that:

$\left|\arctan u-s_{n}(u)\right|\leq\left(\frac{\sqrt{2}\,u}{u+1}\right)^{4n}$

for any $u\in[0,1]$, with $s_{n}$ being an upper bound for $\arctan u$ over
$[0,1]$ for any even $n$ and a lower bound for any odd $n$. If we consider:

$t_{n}(u)=\frac{\pi}{4}-s_{n}\left(\frac{1-u}{1+u}\right)=\frac{\pi}{4}-\sum_{j=0}^{n}\left(-\frac{(1-u)^{4}}{4}\right)^{j}\cdot\left(\frac{1-u}{2(4j+1)}+\frac{(1-u)^{2}}{2(4j+2)}+\frac{(1-u)^{3}}{4(4j+3)}\right),$

then $t_{n}$ is a lower/upper bound for the arctangent function over $[0,1]$
if and only if $s_{n}$ is a lower/upper bound, and:

$\left|\arctan u-t_{n}(u)\right|\leq\left(\frac{1-u}{\sqrt{2}}\right)^{4n}$

holds. Any convex combination of $s_{n}$ and $t_{n}$ is still a lower/upper
bound - by taking:

$w_{n}(u)=\frac{u^{4n+4}\cdot t_{n}(u)+(1-u)^{4n+4}\cdot
s_{n}(u)}{u^{4n+4}+(1-u)^{4n+4}}$

we can perform a reduction of the uniform error, since:

$\left|w_{n}(u)-\arctan u\right|\leq\frac{1}{20^{n}}$

and the error function goes very fast to zero when $u$ approaches $0$ or $1$.
This gives that

$w_{n}\left(\frac{u}{1+\sqrt{1+u^{2}}}\right)$

is an especially good lower/upper bound for the arctangent function when $u$
is close to $0$ or much bigger than $1$, achieving the same uniform error term
with respect to the generalized Shafer-Fink inequality or the continued
fraction expansion.

## References

  * [1] A.M. Fink: Two inequalities, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 6 (1995), 48–49.
(http://pefmath.etf.bg.ac.yu/)

  * [2] Feng Qi, Shi-Qin Zhang, Bai-Ni Guo: Sharpening and generalizations of Shafer’s inequality
for the arc tangent function, Journal of Inequalities and Applications 2009
(2009).

  * [3] R. E. Shafer, E 1867, Amer. Math. Monthly 73 (1966), no. 3, 309.
  * [4] D. S. Mitrinović, Elementary Inequalities, Groningen, 1964.