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# Influence of different mutual friction models on two-way coupled quantized
vortices and normal fluid in superfluid 4He

Hiromichi Kobayashi1 Satoshi Yui2,3 Makoto Tsubota2,3 1Department of Physics &
Research and Education Center for Natural Sciences, Hiyoshi Campus, Keio
University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan  2Department of
Physics, Osaka Metropolitan University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka
558-8585, Japan  3Nambu Yoichiro Institute of Theoretical and Experimental
Physics (NITEP), Osaka Metropolitan University, Osaka 558-8585, Japan

###### Abstract

We study the influence of two mutual friction models on quantized vortices and
normal fluid using two-way coupled simulations of superfluid 4He. The normal
fluid is affected by quantized vortices via mutual friction. A previous study
[Y. Tang, et al. Nat. Commun. 14, 2941 (2023)] compared the time evolutions of
the vortex ring radius and determined that the self-consistent two-way coupled
mutual friction (S2W) model yielded better agreement with the experimental
results than the two-way coupled mutual friction (2W) model whose model
parameters were determined through experiments with rotating superfluid
helium. In this study, we compare the two models in more detail in terms of
the quantized vortex ring propagation, reconnection, and thermal counterflow.
We found that the S2W model exhibits better results than the 2W model on the
microscopic scale near a quantized vortex, such as during quantized vortex
ring propagation and reconnection, although the S2W model requires a higher
spatial resolution. For complex flows such as a thermal counterflow, the 2W
model can be applied even to a low-resolution flow while maintaining the
anisotropic normal fluid velocity fluctuations. In contrast, the 2W model
predicts lower normal fluid velocity fluctuations than the S2W model. The two
models show probability density functions with $-3$ power-law tails for the
normal fluid velocity fluctuations.

##  I Introduction

At temperatures below 2.17 K, superfluid 4He is composed of an inviscid
superfluid component and a viscous normal fluid component. This is known as
the two-fluid model [1, 2]. The circulation of the superfluid component is
quantized and behaves as a quantized vortex line with a diameter of
approximately 10-8 cm. In the hydrodynamics of superfluid 4He, the most
extensively studied experimental phenomenon is the thermal counterflow [3, 4].
In these experiments, two baths are filled with superfluid 4He and connected
via a channel. When one bath is heated, the normal fluid moves into the other
bath through the channel. However, the superfluid moves in the opposite
direction through the channel to satisfy the conservation of mass. As the heat
flux increases, the relative velocity between the superfluid and normal fluid
will also increases. When the relative velocity exceeds a critical value, the
quantized vortices become tangled [5]. This phenomenon is known as quantum
turbulence (QT).

A quantized vortex interacts with a normal fluid through mutual friction in
superfluid 4He [6, 7, 8, 9]. A model of mutual friction [10, 11, 12] was
proposed based on experimental data for uniformly rotating superfluid helium
[13, 14]. This model has been used for one-way coupled simulations, where the
velocity profile of the normal fluid is prescribed, and quantized vortices in
QT are affected via the mutual friction between the superfluid and normal
fluid [15]. Numerical simulations using the vortex filament model (VFM) with
the full Biot-Savart law showed good agreement with the experimental results
[16, 17, 18, 19].

The mutual friction model has also been used in two-way coupled simulations,
where the normal fluid is locally affected by the quantized vortices via
mutual friction. The mutual friction model used in these two-way coupled
simulations is referred to as the 2W model in this study. The 2W model has
been used in two-way coupled simulations with solid boundaries [20, 21],
producing the anomalous anisotropic velocity fluctuations of the normal fluid
in counterflow experiments [22]. However, the undisturbed velocity away from
the core of the quantized vortex is assumed to be the normal fluid velocity in
the 2W model [11].

Two-way coupled simulations using mutual friction with a locally disturbed
normal fluid velocity have also been conducted [23, 24, 25]. The concept of a
self-consistent model [26] was proposed and applied to simple configurations
such as a quantized vortex ring [23] and a straight quantized vortex [27]. The
recent self-consistent model [28] was updated slightly to the present self-
consistent two-way coupled mutual friction model [29], which is referred to as
the S2W model in this study. The S2W model adopts the theoretical friction
force through a vortex line; consequently, no experimentally determined
empirical parameters are required.

The time evolution of a single vortex ring radius obtained using the 2W and
S2W models was compared with the experimental results, and the S2W model
showed better agreement with the experimental results than the 2W model [30].
However, it is necessary to compare the performance of the two models for
flows such as a vortex ring propagation, reconnection, and thermal
counterflow.

Furthermore, monitoring the motion of solid hydrogen tracers in decaying QT
has shown that the probability density function (PDF) of the superfluid
velocity, $v$, is non-Gaussian with $1/v^{3}$ power-law tails owing to the
motion of the quantized vortex [31]. These tails were reproduced in the PDF of
the superfluid velocity obtained with a one-way coupled simulation of the VFM
[32] and the Gross-Pitaevskii equation in a turbulent atomic Bose-Einstein
condensate [33]. It would be interesting to observe the influence of the
superfluid velocity on the PDF of the normal fluid using two-way coupled
simulations with the two models.

In this study, two-way coupled simulations are conducted to examine the
influence of the 2W and S2W models on the velocity fluctuations of a normal
fluid. The remainder of this paper is organized as follows: In Section II, the
basic equations, mutual friction models, and numerical conditions are
described. In Section III, we present and discuss the numerical results for
the vortex ring propagation, reconnection, and thermal counterflow. Finally,
our conclusions are presented in Section IV.

##  II Basic equations and numerical conditions

###  II.1 Basic equations

For the superfluid, the VFM [11, 12, 15] is used to describe the equation of
motion of the quantized vortices. Figure 1 (a) presents a schematic of a
vortex filament. The tangential vector $\bm{s}^{\prime}$ is defined as a unit
vector along the vortex line at point $\bm{s}$. The equation of motion for
point $\bm{s}$ is as follows:

$\frac{d\bm{s}}{dt}=\bm{v}_{s}+\alpha
s^{\prime}\times\bm{v}_{ns}-\alpha^{\prime}\bm{s}^{\prime}\times(\bm{s}^{\prime}\times\bm{v}_{ns}),$
(1)

where $\bm{v}_{s}$ denotes the superfluid velocity,
$\bm{v}_{ns}=\bm{v}_{n}-\bm{v}_{s}$, $\bm{v}_{n}$ is the normal fluid
velocity, and $\alpha$ and $\alpha$’ are the coefficients of mutual friction
at a finite temperature for the 2W model [14, 11, 12].

Figure 1: Schematic of a quantized vortex: (a) vortex filament and tangential
vector $\bm{s}^{\prime}$ at point $\bm{s}$, (b) separated segments with
$\Delta\xi$, and (c) local coupling of a quantized vortex with the normal
fluid discretized on grids. The quantized vortex is coupled with the normal
fluid via mutual friction in the red mesh through which the quantized vortex
passes.

The superfluid velocity at 0 K at position $\bm{r}$ is obtained from the
induced velocity produced from segment $d\bm{s}_{1}$ at position $\bm{s}_{1}$
using the Biot-Savart law as follows:

$\bm{v}_{s}(\bm{r})=\frac{\kappa}{4\pi}\int_{\mathcal{L}}\frac{(\bm{s}_{1}-\bm{r})\times
d\bm{s}_{1}}{|\bm{s}_{1}-\bm{r}|^{3}}+\bm{v}_{s,b}+\bm{v}_{s,a},$ (2)

where $\kappa$ denotes the quantum circulation, $\mathcal{L}$ represents the
integration along the vortex line, and $\bm{v}_{s,b}$ and $\bm{v}_{s,a}$ are
the velocity induced from the boundaries and the uniform flow applied to the
superfluid, respectively.

The momentum equations for the normal fluid are described by the Navier-Stokes
equations:

$\frac{\partial\bm{v}_{n}}{\partial
t}+(\bm{v}_{n}\cdot\nabla)\bm{v}_{n}=-\frac{1}{\rho_{n}}\nabla
P+\nu_{n}\nabla^{2}\bm{v}_{n}+\frac{1}{\rho_{n}}\bm{F}_{ns},$ (3)

where the total density $\rho$ is composed of the superfluid density
$\rho_{s}$ and normal fluid density $\rho_{n}$ as $\rho=\rho_{s}+\rho_{n}$;
$P$ is the effective pressure, $\nu_{n}$ is the kinematic viscosity of the
normal fluid, and $\bm{F}_{ns}$ denotes the mutual friction from the
superfluid to the normal fluid. As shown in Fig. 1 (b), the mutual friction is
calculated from each segment $\Delta\xi$ along the integral path
$\mathcal{L^{\prime}}$ in the normal fluid mesh $\Omega^{\prime}(\bm{r})$
shown in Fig. 1 (c) at position $\bm{r}$ as follows:

$\bm{F}_{ns}=\frac{1}{\Omega^{\prime}(\bm{r})}\int_{\mathcal{L^{\prime}}(\bm{r})}\bm{f}(\xi)d\xi,$
(4)

where $\bm{f}(\xi)$ is the local mutual friction with arc length $\xi$.

###  II.2 2W model

In this study, we compare two models of mutual friction: the 2W model [11, 12]
and the S2W model [29].

First, the 2W model [11, 12] is presented. The mutual friction from the
superfluid to the normal fluid $\bm{f}_{sn}$ is described based on
experimental results [13, 14] as follows:

$\bm{f}_{sn}=-\alpha\rho_{s}\kappa\bm{s}^{\prime}\times\left(\bm{s}^{\prime}\times\bm{v}_{ns}\right)-\alpha^{\prime}\rho_{s}\kappa\bm{s}^{\prime}\times\bm{v}_{ns}.$
(5)

From $\bm{f}(\xi)=\bm{f}_{ns}=-\bm{f}_{sn}$, the local mutual friction in Eq.
(4) can be obtained.

$\bm{f}(\xi)=\rho_{s}\kappa\left[\alpha\bm{s}^{\prime}\times(\bm{s}^{\prime}\times\bm{v}_{ns})+\alpha^{\prime}\bm{s}^{\prime}\times\bm{v}_{ns}\right].$
(6)

The mutual friction $\bm{f}_{sn}$ can be interpreted from another perspective.
The vortex filament is affected by the Magnus force, $\bm{f}_{M}$.

$\bm{f}_{M}=\rho_{s}\kappa\bm{s}^{\prime}\times(\dot{\bm{s}}-\bm{v}_{s}),$ (7)

where $\dot{\bm{s}}=d\bm{s}/dt$. In the 2W model, the drag force $\bm{f}_{D}$
is modeled using the drag coefficients $\gamma_{0}$ and $\gamma_{0}^{\prime}$
[14].

$\bm{f}_{D}=-\gamma_{0}\bm{s}^{\prime}\times\left[\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}})\right]+\gamma_{0}^{\prime}\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}}).$
(8)

Note that $\gamma_{0}$ and $\gamma_{0}^{\prime}$ correspond to $D$ and
$D^{\prime}$, respectively, in reference [11]. In the 2W model, $\bm{f}_{sn}$
satisfies $\bm{f}_{sn}=\bm{f}_{D}$. Because the inertia of the quantized
vortex is negligible, the equation of motion can be expressed as follows:

$\bm{f}_{M}+\bm{f}_{D}=\bm{0}.$ (9)

By eliminating $\dot{\bm{s}}$ using Eq. (1), we obtain the following mutual
friction coefficients:

$\alpha=\frac{\rho_{s}\kappa\gamma_{0}}{\gamma_{0}^{2}+(\rho_{s}\kappa-\gamma_{0}^{\prime})^{2}},\quad\alpha^{\prime}=\frac{\gamma_{0}^{2}-(\rho_{s}\kappa-\gamma_{0}^{\prime})\gamma_{0}^{\prime}}{\gamma_{0}^{2}+(\rho_{s}\kappa-\gamma_{0}^{\prime})^{2}}.$
(10)

###  II.3 S2W model

This section discusses the S2W model [29].

In this model, the drag force is modeled using the drag coefficient $D$ as
follows:

$\bm{f}_{D}=-D\bm{s}^{\prime}\times\left[\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}})\right].$
(11)

When the Reynolds number Revortex of the normal fluid based on the velocity
induced by the vortex line is low ($10^{-5}\sim 10^{-4}$), the coefficient of
the drag force from the vortex line is analytically determined as follows
[34]:

$D=\frac{4\pi\rho_{n}\nu_{n}}{\frac{1}{2}-\gamma-\ln\left(\mathrm{Re}_{vortex}\right)},$
(12)
$\mathrm{Re}_{vortex}=\frac{|\bm{v}_{n\perp}-\dot{\bm{s}}|a_{0}}{4\nu_{n}},$
(13)

where the subscript $\perp$ denotes a component perpendicular to the vortex
line, $\gamma=0.5772$ is the Euler-Mascheroni constant, and $a_{0}$ represents
the vortex core size. In this study, we set $a_{0}$ to $1.3\times 10^{-10}$ m.

In this model, the Iordanskii force [35, 36] is taken into account:

$\bm{f}_{I}=-\rho_{n}\kappa\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}}).$
(14)

The equation of motion results in the following:

$\bm{f}_{M}+\bm{f}_{D}+\bm{f}_{I}=\bm{0}.$ (15)

The mutual friction in this model is $\bm{f}_{sn}=\bm{f}_{D}+\bm{f}_{I}$.
Based on a comparison with Eq. (5), $\bm{f}_{sn}$ is modeled as follows:

$\bm{f}_{sn}=\bm{f}_{D}+\bm{f}_{I}=-\beta\rho_{s}\kappa\bm{s}^{\prime}\times\left(\bm{s}^{\prime}\times\bm{v}_{ns}\right)-\beta^{\prime}\rho_{s}\kappa\bm{s}^{\prime}\times\bm{v}_{ns},$
(16)

where $\beta$ and $\beta^{\prime}$ denote the coefficients of mutual friction
in the S2W model. Note that to ensure $\beta^{\prime}$ is positive, the sign
of $\beta^{\prime}$ is opposite that in reference [29, 39]. The definition of
$\beta^{\prime}$ is consistent with $\alpha^{\prime}$ in Eq. (1) in the 2W
model. Comparing the two mutual friction models in Eq. (8) and Eqs. (11) and
(14), $\gamma_{0}$ and $\gamma_{0}^{\prime}$ correspond to $D$ and
$-\rho_{n}\kappa$, respectively. Substituting these into Eq. (10), the
following mutual friction coefficients are obtained:

$\beta=\frac{\rho_{s}\kappa
D}{D^{2}+(\rho_{s}\kappa+\rho_{n}\kappa)^{2}},\quad\beta^{\prime}=\frac{D^{2}+(\rho_{s}\kappa+\rho_{n}\kappa)\rho_{n}\kappa}{D^{2}+(\rho_{s}\kappa+\rho_{n}\kappa)^{2}}.$
(17)

Finally, we obtain the equation of motion for $\bm{s}$ in the S2W model,
similar to Eq. (1):

$\dot{\bm{s}}=\bm{v}_{s\perp}+\beta\bm{s}^{\prime}\times\bm{v}_{ns}-\beta^{\prime}\bm{s}^{\prime}\times\left(\bm{s}^{\prime}\times\bm{v}_{ns}\right).$
(18)

A summary and comparison of the 2W and S2W models is presented in the
Appendix.

Figure 2 compares the coefficients of the 2W and S2W models as a function of
the temperature, $T$. The total coefficient of the S2W model is larger than
that of the 2W model, and the ratio is stable at approximately two for
temperatures higher than 1.6 K, as shown in Fig. 2 (a) and (b). The fractions
of $\alpha$, $\beta$, $\alpha^{\prime}$, and $\beta^{\prime}$, i.e.,
$\alpha/\sqrt{\alpha^{2}+\alpha^{\prime 2}}$,
$\beta/\sqrt{\beta^{2}+\beta^{\prime 2}}$,
$\alpha^{\prime}/\sqrt{\alpha^{2}+\alpha^{\prime 2}}$, and
$\beta^{\prime}/\sqrt{\beta^{2}+\beta^{\prime 2}}$, respectively, are shown in
Fig. 2 (c) and (d). $\alpha$ is dominant at all temperatures in the 2W model,
whereas $\beta$ decreases gradually with increasing temperature. In contrast,
$\beta^{\prime}$ increases with increasing temperature. These differences
affect the strength of the mutual friction around the quantized vortex, as
discussed in Section III.

Figure 2: Comparison of the coefficients of the 2W and S2W models as a
function of temperature, $T$: (a) total coefficient, (b) coefficient ratio,
(c) coefficient fraction of $\alpha$ and $\beta$, and (d) coefficient fraction
of $\alpha^{\prime}$ and $\beta^{\prime}$. Note that coefficient fraction (c)
$+$ coefficient fraction (d) $\neq 1$.

###  II.4 Numerical methods and conditions

Figure 3: Time evolution of the radius of a quantized vortex ring in the S2W
model for 1.7 K (black), 1.8 K (red) and 2.0 K (green); open symbols:
reference results [39].

Time integration of Eqs. (1) or (18) is performed using the fourth-order
accuracy Runge-Kutta method with $\Delta t=$ 0.0001 s. The spatial resolution
of $\xi$ between discrete points is set to 0.0008 cm $<\Delta\xi<$ 0.0024 cm.
Two filaments are considered to be reconnected if they approach within
$\xi_{min}=$ 0.0008 cm [16]. Short filaments of less than $5\Delta\xi_{min}$
are removed [37].

Equations (3) for the normal fluid are discretized using the second-order
accuracy finite difference method. The simplified Maker and Cell method [38]
is used to couple the velocity and pressure, and the fast Fourier transform is
used to solve the Poisson equation for the pressure.

A temperature, $T$, of 1.9 K is considered. A box size of $D_{x}=D_{y}=D_{z}=$
1 mm is used, and the number of grid points for the normal fluid is set to
$N_{x}=N_{y}=N_{z}=120$.

Periodic conditions are adopted, and the uniform flow
$\bm{v}_{s,a}=-\rho_{n}\bm{V}_{n}/\rho_{s}$ based on the mass conservation law
is applied to the thermal counterflow, where $V_{n}$ is the mean velocity of
the normal fluid. In this study, $V_{n}$ of 2.5 mm/s and 5.0 mm/s in the $x$
direction are considered.

For the quantized vortex ring propagation, the initial radius is set to 0.02
cm. For the reconnection, two quantized vortices cross at a 90-degree angle,
and the initial distance is set to 0.002 cm.

Our implementation of the S2W model is then validated. Figure 3 shows the time
evolution of the radius of a quantized vortex ring using the S2W model for
1.7, 1.8, and 2.0 K. The initial radius is set to $7.6\times 10^{-3}$ cm. The
results show good agreement with the data presented in Ref. [39].

##  III Numerical results and discussion

###  III.1 Quantized vortex ring propagation

Figure 4 shows the quantized vortex ring propagation at 0.05 s for the 2W and
S2W models; the quantized vortex is visualized in red and the normal fluid
vortex tube is displayed in green using the second invariant of the velocity
gradient tensor $Q=0.5$ $s^{-2}$ [40]. The second invariant is defined as
$Q=(W_{ij}W_{ij}-S_{ij}S_{ij})/2$ using the velocity strain tensor
$S_{ij}=(\partial v_{n,j}/\partial x_{i}+\partial v_{n,i}/\partial x_{j})/2$
and the vorticity tensor $W_{ij}=(\partial v_{n,j}/\partial x_{i}-\partial
v_{n,i}/\partial x_{j})/2$. For the 2W model, a pair of rings of normal fluid
vortex tubes are located inside and outside the quantized vortex ring in the
radial direction. In the S2W model, the inner vortex tube remains slightly
behind the quantized vortex ring, and the outer vortex tube propagates
slightly ahead of the quantized vortex ring. This difference between the
models has also been observed at 1.65 K [30]. This is due to the coefficients
of mutual friction of $\alpha$ ($\alpha^{\prime}$) and $\beta$
($\beta^{\prime}$).

Figure 4: Quantized vortex ring propagation at 1.9 K and 0.05 s in the (a) 2W
model and (b) S2W model; red color: quantized vortex, green color: normal
fluid vortex tube ($Q=0.5$ $s^{-2}$). Figure 5: Mutual friction to the vortex
ring $f_{sn}$ on the $x-z$ plane at 1.9 K and 0.05 s in the (a) 2W model
$\bm{f}_{sn}=(-8.62\times 10^{-10},9.50\times 10^{-12})$ N/m and (b) S2W model
with $\bm{f}_{D}=(-3.12\times 10^{-10},-1.69\times 10^{-10})$ N/m and
$\bm{f}_{I}=(-9.25\times 10^{-10},1.71\times 10^{-9})$ N/m where
$\bm{f}_{sn}=\bm{f}_{D}+\bm{f}_{I}=(-1.24\times 10^{-9},1.54\times 10^{-9})$
N/m; red color: quantized vortex; O indicates the center of the quantized
vortex ring.

The mutual friction to the vortex ring $f_{sn}$ on the $x-z$ plane at 0.05 s
is shown in Fig. 5. The mutual friction in the 2W model acts opposite to the
propagation direction of the ring. However, the mutual friction in the S2W
model acts in a diagonal direction. These orientations of the mutual friction
are consistent with those shown in Fig. 2(c) and (d). As the temperature
increases, the direction of the mutual friction in the S2W model rotates from
$-x$ to $z$. The mutual friction in the S2W model is approximately 2.3 times
stronger than that in the 2W model. This result is consistent with the
coefficient ratio in Fig. 2(b). $f_{D}$ is much weaker than $f_{I}$ and acts
inside the ring, whereas $f_{I}$ acts outside the ring, where
$\bm{f}_{sn}=\bm{f}_{D}+\bm{f}_{I}$. Substituting Eq. (18) into Eqs. (11) and
(14) results in the following:

$\bm{f}_{D}=-D\left[(1-\beta^{\prime})\bm{s}^{\prime}\times(\bm{s}^{\prime}\times\bm{v}_{ns})+\beta\bm{s}^{\prime}\times\bm{v}_{ns}\right],$
(19)
$\bm{f}_{I}=\rho_{n}\kappa\left[\beta\bm{s}^{\prime}\times(\bm{s}^{\prime}\times\bm{v}_{ns})-(1-\beta^{\prime})\bm{s}^{\prime}\times\bm{v}_{ns}\right],$
(20)

where

$D=\frac{\rho_{s}\kappa\beta}{\beta^{2}+(1-\beta^{\prime})^{2}},$ (21)
$\rho_{n}\kappa=\frac{\rho_{s}\kappa\left[-\beta^{2}+\beta^{\prime}(1-\beta^{\prime})\right]}{\beta^{2}+(1-\beta^{\prime})^{2}}.$
(22)

It is worth noting that the mutual friction predicted by the S2W model should
approach that predicted by the 2W model at the coarse-graining limit. If
experiments are performed to investigate the location of normal fluid vortex
tubes around a vortex ring, the accuracy of the S2W model will be improved.

The time evolution of the kinetic energy per unit density of the normal fluid
and superfluid is shown in Fig. 6. The kinetic energies of the normal fluid
and superfluid are defined as follows:

$\frac{E_{n}}{\rho_{n}}=\frac{1}{2}v_{n}^{2},\quad\frac{E_{s}}{\rho_{s}}=\frac{1}{2}v_{s}^{2}$
(23)

The energy of the superfluid is gradually decreased by transferring energy to
the normal fluid via mutual friction. The superfluid vortex ring in the 2W
model disappears at 1.0 s, whereas that in the S2W model annihilates at 0.9 s.
The superfluid energy in the 2W model maintains a longer lifetime than that in
the S2W model. This result is consistent with that at 1.65 K [30]. The normal
fluid energy in the S2W model increases faster than that in the 2W model.
After annihilation of the superfluid vortex ring, the normal fluid energy
decreases owing to the viscosity of the normal fluid.

Figure 6: Time evolution of the kinetic energy per unit density of the normal
fluid (solid line) and superfluid (dashed line) at 1.9 K during vortex ring
propagation; black color: 2W model, red color: S2W model.

Next, the PDFs are compared. Figure 7 shows the PDFs of the velocity
fluctuations during the vortex ring propagation. Superfluid velocity
fluctuations are known to have strong non-Gaussian PDFs with $-3$ power-law
tails [31], as shown in Fig. 7 (c) and (d). The superfluid PDFs shown with
solid lines correspond to the event immediately before annihilation of the
quantized vortex rings. No dashed lines are shown because the quantized vortex
ring disappears.

Normal fluid velocity fluctuations are known to have Gaussian PDFs, although
those velocity gradients yield non-Gaussian PDFs [43, 44, 45]. The PDFs of the
normal fluid velocity fluctuations in the $y$ (radial) direction have $-3$
power-law tails, as shown by the fine solid line. The PDFs are affected by the
superfluid fluctuations via mutual friction. The dashed lines indicate the
normal fluid PDFs immediately after the annihilation of the quantized vortex
rings. The PDFs with strong fluctuations are reduced because the superfluid
vortex rings disappear. Figure 7 (a) presents the normal fluid PDFs in the $x$
(propagation) direction. The PDFs have $-3$ power-law tails, which are
affected by the superfluid fluctuations via mutual friction. However, the
tails exist only in the propagation direction, i.e., in the positive $x$
direction. The peaks of the PDFs are located at approximately $-0.5$. This is
due to the normal fluid backflow in the inner region of the quantized vortex
ring. As shown in Fig. 4, the inner normal fluid vortex caused by the
quantized vortex ring rotates through mutual friction and produces the
backflow in the inside of the vortex ring. The backflow is rectified by the
normal fluid contraction flows induced by mutual friction. Consequently, it is
believed that almost no negative fluctuations occur. In terms of the PDFs,
almost no difference between the models is observed.

Figure 7: PDFs of velocity fluctuations at 1.9 K during vortex ring
propagation for the normal fluid in the (a) $x$ (propagation) and (b) $y$
(radial) directions and the superfluid in the (c) $x$ and (d) $y$ directions;
black color: 2W model at 0.9 s (solid line) and 1.1 s (dashed line), red
color: S2W model at 0.8 s (solid line) and 1.0 s (dashed line); solid line:
immediately before annihilation, dashed line: immediately after annihilation;
fine solid line: $1/v^{3}$.

###  III.2 Reconnection of two quantized vortices

Figure 8: Vortex tubes and quantized vortices after reconnection in the (a) 2W
model and (b) S2W model; red color: quantized vortex, green color: normal
fluid vortex tube ($Q=0.5$ $s^{-2}$). Figure 9: Time evolution of the kinetic
energy per unit density of the normal fluid (solid line) and superfluid
(dashed line) during reconnection; black color: 2W model, red color: S2W
model. Figure 10: PDFs of the velocity fluctuations during reconnection for
the normal fluid in the (a) $x$ (quantized vortex propagation) and (b) $y$
(perpendicular) directions and the superfluid in the (c) $x$ and (d) $y$
directions; black color: 2W model at 0.4 s (solid line) and 0.6 s (dashed
line), red color: S2W model at 0.5 s (solid line) and 0.7 s (dashed line);
solid line: immediately before reconnection, dashed line: immediately after
reconnection; fine solid line: $1/v^{3}$. Figure 11: Velocity fluctuations of
the normal fluid in the x (solid line) and y (dashed line) directions during
(a) vortex ring propagation and (b) reconnection; black color: 2W model, red
color: S2W model.

The vortex tubes and quantized vortices after reconnection are shown in Fig.
8; the quantized vortex is shown in red, and the normal fluid vortex tube is
depicted in green using the second invariant $Q=0.5$ s-2 of the velocity
gradient tensor. For the S2W model, the spiral vortex tubes that emerge around
the quantized vortex line have a stronger twist than those in the 2W model.

Figure 9 shows the time evolution of the kinetic energy per unit density of
the normal fluid and superfluid during reconnection. Before reconnection, the
two quantized vortices are twisted while approaching each other owing to the
induced velocity from the other quantized vortex. The motion of the quantized
vortices produces a weak vortex tube of normal fluid via mutual friction.
Consequently, the energy of the superfluid is transferred to the normal fluid.
An abrupt energy transfer from the superfluid to the normal fluid occurs at
approximately 0.5 s. Because the S2W model has stronger mutual friction than
the 2W model, the reconnection is delayed. The normal fluid in the S2W model
has a higher energy than that in the 2W model owing to the stronger mutual
friction.

Figure 10 shows the PDFs of the velocity fluctuations during reconnection. The
PDFs of the normal fluid and superfluid exhibit $-3$ power-law tails in all
directions. This is due to the superfluid fluctuations. Although there is
almost no difference in the superfluid PDFs between the two models, the normal
fluid PDFs drawn with solid lines appear different, as shown in Fig. 10 (a)
and (b). The solid lines correspond to the time immediately before
reconnection. The normal fluid fluctuations remain weak, as shown in Fig. 9
because the influence of the mutual friction on the normal fluid is weak.
Consequently, the PDFs do not yet exhibit $-3$ power-law long tails, i.e., the
PDFs show the results during a transient process. Nevertheless, the two models
produce different PDFs. This is due to the difference in the location of the
normal fluid vortex tubes around the quantized vortex, as shown in Fig. 4.

Figure 11 (a) shows the velocity fluctuations of the normal fluid in the $x$
and $y$ directions during vortex ring propagation. The velocity fluctuation is
defined as the root-mean-square velocity $\sqrt{v_{n,i}^{2}}$ $(i=x,y)$. The
normal fluid in the S2W model receives a greater amount of energy than that in
the 2W model. The velocity fluctuation in the $x$ direction is stronger than
that in the $y$ direction. This is a result of jet formation of the normal
fluid due to the local mutual friction $\bm{f}_{ns}=-\bm{f}_{sn}$ as shown in
Fig. 5. Figure 11 (b) shows the velocity fluctuations of the normal fluid in
the $x$ and $y$ directions during reconnection. As an initial condition, two
straight quantized vortex lines crossed at a 90-degree angle are set at a
certain distance in the $x$ (vertical) direction. Immediately after
reconnection, the 2W model yields a stronger fluctuation in the $x$ direction,
whereas the S2W model yields a stronger fluctuation in the $y$ direction. This
is due to the location of the normal fluid vortex tubes produced around the
quantized vortex line, as shown in Figs. 4 and 8.

###  III.3 Thermal counterflow

This section considers the thermal counterflow. Figure 12 shows a snapshot of
the quantized vortices and normal fluid vortex tubes in a thermal counterflow.
The vortex tubes are produced by mutual friction with quantized vortices. The
S2W model yields a high density of vortex lines and strong vortex tubes. The
vortex line density is defined as the vortex line length per unit volume. The
time evolution of the vortex line density in the thermal counterflow is shown
in Fig. 13. The density gradually increases at 2.5 mm/s, whereas it increases
rapidly at 5.0 mm/s. Subsequently, the density reaches a statistically steady
state. The S2W model yields a density that is approximately twice that of the
2W model. Figure 14 shows the average vortex line density in the statistically
steady state of Fig. 13 as a function of the mean relative velocity, $V_{ns}$.
The slope parameter $\gamma=L^{1/2}/(V_{ns}-v_{0})$ is presented in the
figure, where $v_{0}$ is a small, adjustable parameter on the order of 1 cm
s-1. The value $\gamma=167$ of 2W∗ is the reference value in our previous
study using the 2W model [22] and is consistent with the present result of
$\gamma=160$ obtained using the 2W model. The S2W model yields a higher
$\gamma=187$ than the 2W model, although the experimental value is $\gamma\sim
130$ s/cm2 [41, 42].

The Vinen equation [4, 8] can be expressed as follows:

$\frac{dL}{dt}=A|V_{ns}|L^{3/2}-BL^{2},$ (24)

where $A$ and $B$ denote the coefficients of the generation and decay rates,
respectively. In the steady state, $L^{1/2}=A/B|V_{ns}|$ is obtained. Here,
$A/B\sim\gamma$. As shown in Fig. 9, the stronger mutual friction in the S2W
model leads to a slower approach between two quantized vortices. Consequently,
the decay rate becomes low, and thus the S2W model estimates a larger $\gamma$
than the 2W model.

Figure 12: Quantized vorticies (red) and normal fluid vortex tubes (green) in
a thermal counterflow for the 2W model (a) $v_{n}=2.5$ mm/s ($Q=10$ $s^{-2}$),
(b) $v_{n}=5.0$ mm/s ($Q=100$ $s^{-2}$) and S2W model (c) $v_{n}=2.5$ mm/s
($Q=100$ $s^{-2}$), (d) $v_{n}=5.0$ mm/s ($Q=1000$ $s^{-2}$). Figure 13: Time
evolution of the vortex line density in the thermal counterflow for the 2W
model and S2W model at $v_{n}=2.5$ and 5.0 mm/s. Figure 14:  Average vortex
line density in the statistically steady state of Fig. 13 as a function of the
mean relative velocity, $V_{ns}$, where $\gamma=L^{1/2}/(V_{ns}-v_{0})$ and
$v_{0}$ is a small, adjustable parameter on the order of 1 cm s-1; 2W∗ denotes
the result in our previous study using the 2W model [22].

The PDFs of the velocity fluctuations in the thermal counterflow in the
statistically steady state are shown in Fig. 15. Two mean normal fluid
velocities of 2.5 mm/s and 5.0 mm/s are examined. There is no difference in
the direction of fluctuations in the PDFs of the superfluid. The PDFs of the
normal fluid in the $x$ and $y$ directions exhibit Gaussian distributions with
weak fluctuations. As shown in Fig. 15 (b), strong fluctuations exhibit $-3$
power-law tails. The S2W model yields stronger fluctuations than the 2W model
owing to the stronger mutual friction. The mean normal fluid velocity has a
weak influence on the intensity of the PDFs of the normal fluid. The PDFs in
the $x$ (streamwise) direction appear asymmetric in Fig. 15 (a). The positive
fluctuations exhibit sub-Gaussian distributions, whereas the negative
fluctuations produce $-3$ power-law tails. In this study, the normal fluid
flows in the positive $x$ direction, and the superfluid moves in the negative
$x$ direction during the counterflow. If quantized vortex rings are generated,
they tend to propagate in the negative $x$ direction. Consequently, long-tail
PDFs are produced in the propagation direction of the quantized vortex rings,
as shown in Fig. 7 (a).

Figure 15: PDFs of velocity fluctuations in the thermal counterflow for the
normal fluid in the (a) $x$ (streamwise) and (b) $y$ (perpendicular)
directions and the superfluid in the (c) $x$ and (d) $y$ directions; black
color: 2W model, red color: S2W model; solid line: mean normal fluid velocity
of 2.5 mm/s, dashed line: 5.0 mm/s; fine solid line: $1/v^{3}$ and Gaussian
profile for (a) and (b).

Figure 16 shows the normal fluid velocity fluctuations in the 2W and S2W
models as a function of the mean normal fluid velocity in the thermal
counterflow. At a low resolution, the 2W model reproduces the anisotropic
fluctuations, whereas the S2W model produces fewer anisotropic fluctuations.
At high resolution, the 2W model yields almost the same fluctuations as the
low-resolution model, whereas the S2W model generates anisotropic
fluctuations. The S2W model requires higher resolution and yields higher
fluctuations than the 2W model. However, the S2W model predicts the normal
fluid fluctuations better than the 2W model.

Figure 16: Normal fluid velocity fluctuations in the $x$ (streamwise, solid
line) and $y$ (perpendicular, dashed line) directions as a function of the
mean normal fluid velocity, $V_{n}$, in the thermal counterflow for grid
numbers of (a) $40^{3}$ and (b) $120^{3}$; Exp: experimental results (blue)
[46], 2W model (black), and S2W model (red).

##  IV Conclusions

We investigated the performance of two different mutual friction models, i.e.,
the 2W and S2W models, on quantized vortices and normal fluid using two-way
coupled simulations of superfluid 4He. In the quantized vortex ring
propagation and reconnection, the normal fluid vortex tube induced by mutual
friction is produced at slightly different locations around the quantized
vortex in each model. The normal fluid velocity fluctuations in the S2W model
are stronger than those in the 2W model, whereas the probability density
functions produced by the two models show negligible differences.

The S2W model is better suited for describing the normal fluid physics on a
microscopic scale near a quantized vortex, such as during quantized vortex
ring propagation and reconnection. For complex flows, such as a thermal
counterflow, the 2W model can represent a low-resolution flow while
maintaining anisotropic fluctuations. However, the S2W model requires higher
resolution and yields higher fluctuations than the 2W model. The two-way
coupled simulation with each model produces PDFs with $-3$ power-law tails for
the normal fluid velocity fluctuations.

###### Acknowledgements.

H. K. acknowledges the support from JSPS KAKENHI (Grant Number JP22H01403). S.
Y. acknowledges the support from JSPS KAKENHI (Grant Number JP23K13063). M. T.
was supported by JSPS KAKENHI (Grant Numbers JP22H05139 and JP23K03305).

*

##  Appendix A Summary of the 2W and S2W models

We summarize and compare the parameters, forces, and equations of motion of
the 2W and S2W models in Table 1.

Table 1: Comparison of the 2W model and S2W model 2W model | S2W model
---|---
$\alpha=\frac{\rho_{s}\kappa\gamma_{0}}{\gamma_{0}^{2}+(\rho_{s}\kappa-\gamma_{0}^{\prime})^{2}}$ | $\beta=\frac{\rho_{s}\kappa D}{D^{2}+(\rho_{s}\kappa+\rho_{n}\kappa)^{2}}$
$\alpha^{\prime}=\frac{\gamma_{0}^{2}-(\rho_{s}\kappa-\gamma_{0}^{\prime})\gamma_{0}^{\prime}}{\gamma_{0}^{2}+(\rho_{s}\kappa-\gamma_{0}^{\prime})^{2}}$ | $\beta^{\prime}=\frac{D^{2}+(\rho_{s}\kappa+\rho_{n}\kappa)\rho_{n}\kappa}{D^{2}+(\rho_{s}\kappa+\rho_{n}\kappa)^{2}}$
$\gamma_{0}=\frac{\rho_{s}\kappa\alpha}{\alpha^{2}+(1-\alpha^{\prime})^{2}}$ | $D=\frac{\rho_{s}\kappa\beta}{\beta^{2}+(1-\beta^{\prime})^{2}}$
$\gamma_{0}^{\prime}=\frac{\rho_{s}\kappa\left[\alpha^{2}-\alpha^{\prime}(1-\alpha^{\prime})\right]}{\alpha^{2}+(1-\alpha^{\prime})^{2}}$ | $\rho_{n}\kappa=\frac{\rho_{s}\kappa\left[-\beta^{2}+\beta^{\prime}(1-\beta)\right]}{\beta^{2}+(1-\beta^{\prime})^{2}}$
$\bm{f}_{M}=\rho_{s}\kappa\bm{s}^{\prime}\times(\dot{\bm{s}}-\bm{v}_{s})$ | $\bm{f}_{M}=\rho_{s}\kappa\bm{s}^{\prime}\times(\dot{\bm{s}}-\bm{v}_{s})$
$\bm{f}_{D}=-\gamma_{0}\bm{s}^{\prime}\times\left[\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}})\right]+\gamma_{0}^{\prime}\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}})$ | $\bm{f}_{D}=-D\bm{s}^{\prime}\times\left[\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}})\right]$
| $\bm{f}_{I}=-\rho_{n}\kappa\bm{s}^{\prime}\times(\bm{v}_{n}-\dot{\bm{s}})$
$\bm{f}_{M}+\bm{f}_{D}=\bm{0}$ | $\bm{f}_{M}+\bm{f}_{D}+\bm{f}_{I}=\bm{0}$
$\bm{f}_{sn}=\bm{f}_{D}$ | $\bm{f}_{sn}=\bm{f}_{D}+\bm{f}_{I}$
$\bm{f}_{sn}=-\alpha\rho_{s}\kappa\bm{s}^{\prime}\times\left(\bm{s}^{\prime}\times\bm{v}_{ns}\right)-\alpha^{\prime}\rho_{s}\kappa\bm{s}^{\prime}\times\bm{v}_{ns}$ | $\bm{f}_{sn}=-\beta\rho_{s}\kappa\bm{s}^{\prime}\times\left(\bm{s}^{\prime}\times\bm{v}_{ns}\right)-\beta^{\prime}\rho_{s}\kappa\bm{s}^{\prime}\times\bm{v}_{ns}$
$\dot{\bm{s}}=\bm{v}_{s}+\alpha s^{\prime}\times\bm{v}_{ns}-\alpha^{\prime}\bm{s}^{\prime}\times(\bm{s}^{\prime}\times\bm{v}_{ns})$ | $\dot{\bm{s}}=\bm{v}_{s\perp}+\beta\bm{s}^{\prime}\times\bm{v}_{ns}-\beta^{\prime}\bm{s}^{\prime}\times\left(\bm{s}^{\prime}\times\bm{v}_{ns}\right)$

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# The SERENDIP III 70 cm Search for Extraterrestrial Intelligence

Stuart Bowyer, Michael Lampton, Eric Korpela, Jeff Cobb, Matt Lebofsky, and
Dan Werthimer  Space Sciences Laboratory, University of California, Berkeley,
CA 94720

###### Abstract

We employed the SERENDIP III system with the Arecibo radio telescope to search
for possible artificial extraterrestrial signals. Over the four years of this
search we covered 93% of the sky observable at Arecibo at least once and 44%
of the sky five times or more with a sensitivity of $\sim 3\times
10^{-25}Wm^{-2}$. The data were sent to a $4\times 10^{6}$ channel spectrum
analyzer. Information was obtained from over $10^{14}$ independent data points
and the results were then analyzed via a suite of pattern detection algorithms
to identify narrow band spectral power peaks that were not readily
identifiable as the product of human activity. We separately selected data
coincident with interesting nearby G dwarf stars that were encountered by
chance in our sky survey for suggestions of excess power peaks. The peak power
distributions in both these data sets were consistent with random noise. We
report upper limits on possible signals from the stars investigated and
provide examples of the most interesting candidates identified in the sky
survey. This paper was intended for publication in 2000 and is presented here
without change from the version submitted to ApJS in 2000.

##  I. Introduction

Early radio searches for extraterrestrial intelligence used dedicated
telescope time to search for emission from nearby stars (see Tarter 1991 for a
partial listing, and Tarter 2001 for a full listing of these searches). This
type of search became increasingly difficult to carry out at major facilities
because of a general reluctance to devote dedicated telescope time to such
projects, which though interesting, are acknowledged to have a low probability
of success. In addition, sky surveys were carried out which scanned
substantial portions of the sky. The Berkeley SERENDIP project (Search for
Extraterrestrial Radio Emissions from Nearby Developed Intelligent
Populations) solved the dedicated telescope time problem by using data
obtained simultaneously with ongoing astronomical research. This program began
over twenty years ago (Bowyer et al. 1983) and has continued to the present
day with ever increasing sensitivity and an ever-widening set of search
parameters.

Other sky surveys using dedicated telescopes have been, and are continuing to
be, carried out. The Ohio State program (Dixon 1985) was the earliest sky
survey; this project is now terminated. The Harvard search (Leigh & Horowitz
1997) has also been terminated. The Argentinian search (Lemarchand et al.
1997), and the Australian search (Stootman et al. 1999) continue. Targeted
searches of nearby stars have been initiated by the SETI Institute (Tarter
1997) using substantial amounts of dedicated telescope time which were
obtained in return for a substantial financial contribution to the telescope
upgrade which was carried out after the conclusion of the SERENDIP III
observations.

We discuss our sky survey search for artificial extraterrestrial signals with
the SERENDIP III system (Bowyer et al. 1997) and the Arecibo telescope.
Although the search was a sky survey, nearby solar-type stars inevitably fell
within the beam pattern of the telescope in the course of these observations.
As part of the analysis of the SERENDIP III data, we have separately
investigated the data from observations of the sky coincident with these
nearby stars. Although our integration times for individual targets are
relatively short (as compared, for example, with the SETI Institute targeted
search), our sensitivity is still substantial because of the large collecting
area of the Arecibo telescope and the outstanding receivers that are available
for use with this instrument.

We report the results of our sky survey and provide upper limits on possible
signals from stars in this paper.

##  II. Observations and Analysis

The SERENDIP III data were obtained with the National Astronomy Ionosphere
Center’s radio observatory in Arecibo, Puerto Rico with a 430 MHz receiver.
Data collection began in April, 1992 and ended in October, 1998.

The feed used for SERENDIP III was located at the opposite carriage house from
the primary observer’s feed. This resulted in three observation modes. In the
first, representing about 45% of the observing time, the primary observer’s
feed was tracking a point on the sky. This resulted in the SERENDIP III beam
moving across the sky at roughly twice the sidereal rate. In the second,
representing about 40% of the observing time the feeds were stationary,
resulting in motion at the sidereal rate. In the third, representing about 15%
of the observing time, the SERENDIP III feed was tracking a point on the sky.

Figure 1.— An illustration for the SERENDIP III data flow showing the steps
taken to detect candidate signals. Real-time processing consists of power
spectra generation and application of an adaptive threshold to the resulting
spectra. Off-line data reduction included RFI rejection techniques, pattern
detection, and candidate extraction.

Power spectra were generated by the SERENDIP III four million channel spectrum
analyzer which had a 1.7 second integration period and a 0.6 Hz frequency
resolution (2.5 MHz instantaneous band coverage). The sensitivity achieved in
a 1.7 second integration at our typical system temperature of 45 K was
$3\times 10^{-25}$ W m-2. This bin size is wide enough to encompass Doppler
frequency drifts caused by the Earth’s motions plus reasonable accelerations
of a hypothetical transmitter’s reference frame. The receiver’s entire 12 MHz
band was processed in 2.4 MHz steps taking about 8.5 seconds to complete a
single sweep. The SERENDIP III fast Fourier transform based hardware is
described in detail by Werthimer, et al. (1997).

Adaptive thresholding was achieved by baseline smoothing the raw power spectra
with a sliding eight thousand channel local-mean boxcar and searching for
channels exceeding 16 times the mean spectral power. These signals were
recorded along with time, pointing coordinates, detection frequency, and
signal power.

Following the real-time data analysis by the SERENDIP III instrument, the
reduced data was shipped to Berkeley of off-line data reduction and candidate
generation. Off-line data reduction included radio frequency interference
(RFI) rejection techniques, pattern detection, and candidate extraction. The
overall data flow is shown in Figure 1.

###  II.1. Off-line Data Reduction

Data were transferred from Arecibo to Berkeley across the Internet where off-
line data analysis activities began. Data reduction consisted of removing data
taken during periods when the telescope was slewing too fast or too slow for
our analysis procedures, followed by the application of a suite of RFI
filters.

Excessively rapid telescope slew rates precluded acquisition of accurate
positioning information. In addition, during times when the receiver tracks a
point on the sky, it is not possible for our analysis programs to
differentiate between continuous RFI and a potential signal. Therefore, our
first filter was to censor data acquired during periods of rapid telescope
movement and periods when the telescope was tracking sky objects. Roughly 15%
of the data were removed for this reason.

The next step in data reduction was non-drifting RFI rejection. SERENDIP’s
non-drifting RFI rejection algorithms incorporated dynamically adaptive
statistical analysis routines that detect spurious signals from terrestrial
and near-space sources. Three cluster analysis tests were conducted on each
input data file spanning several hours of observation. Signals were rejected
if they (1) were detected over broad areas of the spectrum in one or more
integration periods (broad spectrum interference), (2) persisted at the same
receiver frequency through multiple telescope beams, or (3) persisted in the
same channel of the spectrum analyzer.

Broad spectrum interference was identified by the rejection test:

$x>50\hskip 7.22743ptand\hskip 7.22743pt\frac{\sum{d^{2}}}{x}>S$ (1)

where d is the number of frequency bins (0.6 Hz/bin) between simultaneous
events above the threshold and $x$ is the number of events above threshold in
the spectrum. For SERENDIP III, the threshold value was set at S = $10^{8}$.

RFI rejection algorithm (2) uses a statistical method to determine if several
detections at the same observing frequency could be ruled out. If these
detections occur with a significantly above-average hit rate, and they
continue to occur when our observing beam has moved beyond one beam width
(0.17 degrees), we reject the hypothesis of random Poisson events being the
cause. Instead we mark these detections as being due to external RFI, and
reject them.

RFI rejection algorithm (3) uses the same statistical test as (2) but applies
it to hit sequences that have the same intermediate-frequency bin number. In
this way it rejects interference generated within the observatory.

Data surviving the first three rejection criteria were further analyzed for
RFI that drifts rapidly in frequency and were therefore not rejected by
algorithms (2) and (3) above. Figure 2 illustrates SERENDIP’s drifting
frequency RFI detection algorithm.

Figure 2.— Schematic illustration of SERENDIP’s drifting RFI detection
algorithm. Sectors of frequency-time space around each detection were analyzed
for signals that exhibit time-coherency and persistence over multiple
pointings of the telescope’s beam.

The average event density in frequency-time space was calculated for each data
set. Sectors in frequency-time space around each detection (shown as dots in
Figure 2) were analyzed to identify sectors having an unusually high number of
detections. Sectors containing an excessive number of detections were further
analyzed for the presence of drifting signals that persisted through multiple
pointings of the telescope beam.

An example of the efficacy of the RFI rejection techniques is shown in Figure
3. Note that after application of SERENDIP’s RFI filters, the remaining data
closely approached the number expected from a Poisson noise distribution.
Typically, 98% of the RFI was rejected, yet only 1% of the band was lost
during RFI removal.

Figure 3.— Data from a period of severe interference (top). Each dot
represents a high power signal detection by the SERENDIP instrument. After RFI
filtering (bottom) the number of hit remaining approaches that expected from a
Poisson distribution, as would be expected in a white-noise environment.

###  II.2. Pattern Detection

A suite of pattern detection algorithms was employed to detect signal
recurrence and telescope beam pattern matching. The primary pattern looked for
was recurrence of a signal in frequency over time and position. All data were
frequency corrected to the solar system barycenter. Thus, only those signals
that displayed the proper Doppler drift relative to the solar system
barycenter remained at a nearly constant frequency. If a signal recurred
within a defined frequency window, it was tagged as a candidate. We defined
frequency windows of two different widths in order to find two different
classes of signals.

The first class (Class I) consists of signals with no frequency drift in the
barycentric frame. If transmitted from a planet, such a signal must contain a
Doppler correction, and therefore would be a deliberately beamed message. For
such signals, the chief cause of potential frequency drift arises from our
lack of knowledge of the source’s exact position on the sky. We use the
direction of the beam center to approximate the source position, and obtain an
approximate compensating frequency drift by projecting our antenna’s diurnal
and annual acceleration onto that nominal line of sight. Due to our beam’s
size (0.17 degrees), this drift rate uncertainty can shift an ideal signal by
as much as 120 Hz. Consequently, our compensated signal class defined a
candidate as any set of signals that recurred within a band whose width
$f_{\sigma}=$ 120 Hz.

The second class (Class II) consists of signals containing significant Doppler
acceleration. Radio leakage from a transmitter on a planet will significantly
drift in frequency over time spans of minutes to months. Here, signal sets
were allowed a frequency deviation $f_{\sigma}=$ 50 kHz. The width of this
band was derived assuming a transmitter whose acceleration was as great as the
Jovian cloud tops.

Multiple detections in a given transit were examined to determine if the
observed power as a function of time matched the telescope’s Gaussian beam
pattern. Such a finding would suggest that the signal emanated from a point
source on the sky.

###  II.3. Candidate Merging and Ranking

The output of SERENDIP’s suite of pattern detection algorithms was a
collection of statistically interesting events. These data sets were then
examined to identify candidate locations for signals of interest. A candidate
was defined as a one beamwidth area on the sky that was identified by one or
more pattern detection algorithms. A candidate is fully described by one or
more records in the candidate database, each of which is the result of
detection by a single algorithm. In addition, celestial objects such as nearby
stars, globular clusters, and known planetary systems are entered into the
candidate database to check for coincidence with candidates identified by the
other algorithms described herein.

Candidates are ranked by two methods. First, each candidate record was given
an algorithm specific score. Second, we asked how many algorithms detected the
candidate. Here we considered coincidence with an interesting celestial object
to be counted as an “algorithm”.

To determine the score in the case of multiple transit detections, we
calculated the relative probability of our detections occurring from noise,
given the number of detections in the candidate area, the number of times we
have observed the candidate area, the number of frequency bins searched by the
algorithm, and the actual frequency separation of the detections. The relative
probability is given by

$P=\frac{n^{n_{d}}_{e}}{{n_{d}}{!}{\left(\frac{F_{tot}}{F_{win}}\right)}^{({n_{d}}-1)}}\left[\frac{\Delta{f}+f_{\sigma}}{f_{\sigma}}\right]$
(2)

where:

$n_{e}$ is the total number of events logged at any frequency in the candidate
area,

$n_{d}$ is the number of detections within $F_{win}$,

$F_{win}$ is the frequency window searched for events,

$F_{tot}$ is the total SERENDIP band observed,

$\Delta{f}$ is the maximum frequency separation for the detection set,

$f_{\sigma}$ is the expected frequency variance as explained in section 2.3.

In searches for either Class I or Class II signals, the lower the numerical
rating in this algorithm, the more promising the candidate.

To calculate the score in the case of beam pattern matching in a single
transit across the telescope’s Gaussian beam, we first determined the best fit
Gaussian to the data. The relative score is then given by

$P_{G}=A_{G}\hskip 3.61371ptQ(\chi^{2}\hskip 3.61371pt|\hskip 3.61371pt\nu)$
(3)

where $A_{G}$ is the amplitude of the best fit Gaussian, and $Q(\chi^{2}\hskip
3.61371pt|\hskip 3.61371pt\nu)$ is the $\chi^{2}$ probability function. In
this case, the higher the numerical rating, the more promising the candidate.

The data from the most promising candidates were independently scrutinized for
RFI contamination by three researchers and those surviving this evaluation
constitute the list of most interesting candidates. Our top candidates from
our signal detection algorithms and their scores are given in Table 1.

Table 1Most Interesting Candidates R.A. | Dec | Sky | Frequency | Max. Freq. | # of | # of | Score by | Coincident
---|---|---|---|---|---|---|---|---
|  | Separation |  | Separation11Maximum frequency separation between multiple observations | Obs.22Number of times SERENDIP III observed this point in the sky | Times | Algorithm33Three different scoring algorithms:
Algorithm 1: Doppler compensated signal class (120Hz max frequency window)
(see eq. 2); smaller is more remarkable
Algorithm 2: Non-Doppler compensated signal class (50kHz max frequency window)
(see eq. 2); smaller is more remarkable
Algorithm 3: Gaussian beam fit (see eq. 3); larger is more remarkable | Object
(hours) | (deg) | (deg) | (MHz) | (Hz) |  | Detected | 1 | 2 | 3 |
0.67 | 28.1 | 0.1 | 429.600 | 6 | 6 | 2 | 91.084 |  |  | GJ 1019
0.96 | 7.8 | 0.1 | 426.839 | - | 1 | 1 |  |  | 43.17 |
2.13 | 26.9 | 0.1 | 431.959 | 17 | 20 | 3 | 1.801 |  |  |
4.59 | 27.0 | 0.1 | 434.603 | 30 | 13 | 3 | 0.632 |  |  |
5.13 | 2.1 | 0.2 | 435.009 | 46467 | 4 | 2 |  | 79.371 | 22.68 |
8.15 | 9.0 | 0.1 | 435.089 | - | 1 | 1 |  |  | 26.93 | GJ 299
10.23 | 18.5 | 0.2 | 430.059 | 79 | 34 | 3 | 20.525 |  |  |
14.32 | 25.9 | 0.2 | 427.350 | 72 | 8 | 3 | 0.250 |  |  |
23.09 | 26.8 | 0.1 | 429.973 | 27 | 21 | 3 | 2.529 |  |  |

###  II.4. Investigation of Possible Signals from Nearby G-Dwarf Stars

We investigated the data obtained from nearby G-dwarf stars that were observed
by chance in our sky survey with the rationale that these were especially
interesting candidates. In this effort, we used the Center for Astrophysics
list of G-dwarf stars within (roughly) 100pc of the Sun (Latham 1999) as
listed on their web site as of August 1999111http://cfa-
www.harvard.edu/$\sim$latham/gdwarf.html. We identified those stars in this
list visible with the Arecibo telescope (i.e. stars between $-2^{\circ}$
and$+38^{\circ}$). We then searched the SERENDIP III position data set and
identified stars in the list that fell within the half power beam width of the
Arecibo antenna during our observations.

Of the 516 stars in the CFA list that are observable with the Arecibo
telescope, 494 had been observed at least once, and 439 had been observed
multiple times. In Figure 4 we show a histogram of the number of times these
stars were observed.

Figure 4.— A histogram of the number of separate times each star in the CFA
G-dwarf subset was observed in the SERENDIP III search. No narrow band excess
power beyond that expected for random noise was detected from these stars.

We derived upper limits to the excess power from these stars by first checking
the most recent version of the Hipparcos catalogue for the stellar distances.
The number of stars observed versus distance is shown in Figure 5. We then
derived the upper limits to the source stellar luminosity using the Hipparcos
distances and the observed fluxes. We note that the sensitivity limits are
uncertain to a factor of two because we used the average sensitivity of the
antenna rather than the sensitivity at the exact position of the star within
the beam.

Figure 5.— A histogram showing the Hipparcos distance distribution of stars in
the CFA G-dwarf subset.

In Table 2 we provide the HD number of the observed stars, their distances as
obtained from the Hipparcos catalogue, and the upper limits to the irradiated
power at the star.

##  III. Conclusions

We have carried out an extensive search with the world’s largest telescope for
evidence of radio emission produced by an extraterrestrial intelligence. We
were able to carry out this search using this unique facility because of the
non-intrusive character of our observing program. A major challenge in our
search (and in all other searches for extra terrestrial intelligence) is the
problem of false signals produced as a result of human activity. We developed
a variety of techniques to deal with this problem and demonstrated their
robustness. This shows that a non-intrusive collateral data collection
technique such as ours is viable.

Given the extensive character of our search we found many signals of potential
interest. A prioritization scheme was developed to identify the most promising
of these signals. These were examined in more detail. In the end, no
extraterrestrial signals were identified. We do not find this surprising nor
discouraging given our lack of knowledge as to appropriate source locations,
frequencies and time periods that an intentional extraterrestrial signal may
be employing.

We are continuing our search.

We thank the Arecibo staff for their support in carrying out this project
which they did as an after-hours volunteer project. The Arecibo Observatory is
part of the National Astronomy and Ionosphere Center which is operated by
Cornell University under a Cooperative Agreement with the National Science
Foundation. We acknowledge the efforts of the many volunteers who have worked
on SERENDIP over the years. We acknowledge a generous gift by Watson and
Marilyn Alberts to carry out this analysis of the SERENDIP data. This analysis
has also been supported by grants from the SETI Institute and the Planetary
Society. The SERENDIP III project was supported by significant contributions
from Mrs. Elizabeth Bowyer, the Planetary Society, and the Toshiba
Corporation. Additional support was provided by Intel, Xilinx, and the Friends
of SERENDIP.

## References

  * Bowyer et al. (1983) Bowyer, S., Zeitlin, G., Tarter, J., Lampton, M., & Welch, W 1983, Icarus, 53, 147
  * Bowyer et al. (1997) Bowyer, S., Werthimer, D., Donnelly, C., Cobb, J., Ng, D., & Lampton, M. 1997, in Astronomical and Biochemical Origins and the Search for Life in the Universe, ed. C. Cosmovici, S. Bowyer and D. Werthimer (Bologna: Editrice Compositori), 667
  * Cobb et al. (1997) Cobb, J., Donnelly, C., Bowyer, S., Werthimer, D., & Lampton, M., 1997, in Astronomical and Biochemical Origins and the Search for Life in the Universe, ed. C. Cosmovici, S. Bowyer and D. Werthimer (Bologna: Editrice Compositori), 677
  * Dixon (1985) Dixon, R. 1985, in The Search for Extraterrestrial Life: Recent Developments, ed. M. Papagiannis (Dordrecht: D. Reidel Publishing Co.), 305
  * Leigh & Horowitz (1997) Leigh, D. & Horowitz, P. 1997, in Astronomical and Biochemical Origins and the Search for Life in the Universe, ed. C. Cosmovici, S. Bowyer and D. Werthimer (Bologna: Editrice Compositori), 601
  * Latham (1999) Latham, D. 1999, in Bioastronomy 99: A New Era in Bioastronomy, ed. G Lemarchand and K. Meech (Astronomical Society of the Pacific), 19
  * Lemarchand et al. (1997) Lemarchand, G., Colomb, F. R., Hurrell, E. E., & Olalde, J. C. 1997, in Astronomical and Biochemical Origins and the Search for Life in the Universe, ed. C. Cosmovici, S. Bowyer and D. Werthimer (Bologna: Editrice Compositori), 611
  * Stootman de Horta Wellington & Oliver (1999) Stootman, F. H., de Horta, A. Y., Wellington, K. J. & Oliver, C. 1999, in Bioastronomy 99: A New Era in Bioastronomy, ed. G Lemarchand and K. Meech (Astronomical Society of the Pacific), 90
  * Tarter (1991) Tarter, J. C. 1991, in Frontiers of Life, Third Recontres de Blois, ed. J. and K. Tran Tranh Van, J. C. Mounolou, J. Schneider, C. McKay (France: Editions Frontieres), 351
  * Tarter (1997) Tarter, J. C. 1997, in Astronomical and Biochemical Origins and the Search for Life in the Universe, ed. C. Cosmovici, S. Bowyer and D. Werthimer (Bologna: Editrice Compositori), 633
  * Tarter (2001) Tarter, J. C. 2001, ARA&A, in preparation
  * Werthimer et al. (1997) Werthimer, D., Bowyer, S., Ng, D., Donnelly, C., Cobb, J., Lampton, M., & Airieau, S. 1997, in Astronomical and Biochemical Origins and the Search for Life in the Universe, ed: C. Cosmovici, S. Bowyer and D. Werthimer (Bologna: Editrice Compositori), 683

Table 2Stars observed with SERENDIP III HD number | Distance | Upper Limit EIRP
---|---|---
| (pc) | (log10(Watts))
101 | 38 | 12.7
377 | 40 | 12.8
531E | 70 | 13.2
531W | 70 | 13.2
700 | 54 | 13.0
1832 | 41 | 12.8
4903 | 53 | 13.0
5035 | 39 | 12.7
6715 | 33 | 12.6
7047A | 40 | 12.7
8009A | 65 | 13.2
8009B | 65 | 13.2
8262 | 26 | 12.4
8523 | 60 | 13.1
8574 | 44 | 12.8
8648 | 45 | 12.9
9091 | 46 | 12.9
9224 | 43 | 12.8
9472 | 33 | 12.6
9670 | 37 | 12.7
9986 | 26 | 12.4
10126 | 34 | 12.6
10844 | 54 | 13.0
11045 | 58 | 13.1
11130 | 27 | 12.4
11850 | 33 | 12.6
12235 | 31 | 12.5
12783 | 48 | 12.9
13357A | 49 | 12.9
13357B | 49 | 12.9
13382 | 33 | 12.6
13483 | 34 | 12.6
13836 | 37 | 12.7
13997 | 34 | 12.6
14082B | 39 | 12.7
14305 | 34 | 12.6
14348 | 62 | 13.1
14651 | 40 | 12.8
14874 | 59 | 13.1
15632 | 42 | 12.8
16086 | 54 | 13.0
16397 | 36 | 12.7
17674 | 47 | 12.9
17820 | 65 | 13.2
18144 | 26 | 12.4
18330 | 39 | 12.7
18702 | 32 | 12.6
18774 | 66 | 13.2
19019 | 23 | 12.3
19308A | 43 | 12.8
19445 | 39 | 12.7
19518 | 41 | 12.8
19902 | 42 | 12.8
19962 | 38 | 12.7
20165 | 22 | 12.2
20477 | 52 | 13.0
21183 | 75 | 13.3
21663A | 46 | 12.9
21774 | 50 | 13.0
22309 | 45 | 12.9
23314 | 49 | 12.9
24040 | 47 | 12.9
24053 | 33 | 12.6
24206 | 179 | 14.1
24496 | 21 | 12.2
24552 | 45 | 12.9
24702 | 47 | 12.9
25295 | 65 | 13.2
25682 | 46 | 12.9
25825 | 47 | 12.9
26749 | 36 | 12.7
26756 | 46 | 12.9
26767 | 45 | 12.9
26913 | 21 | 12.2
26923 | 21 | 12.2
27282 | 47 | 12.9
27406 | 45 | 12.9
27642 | 62 | 13.1
27732 | 48 | 12.9
27771 | 47 | 12.9
27859 | 48 | 12.9
28099 | 47 | 12.9
28205 | 46 | 12.9
28237 | 47 | 12.9
28258 | 47 | 12.9
28344 | 47 | 12.9
28406B | 46 | 12.9
28462 | 40 | 12.8
28474 | 53 | 13.0
28580 | 36 | 12.7
28676 | 39 | 12.7
28992 | 43 | 12.8
29150 | 34 | 12.6
29356 | 63 | 13.1
29461 | 48 | 12.9
30101E | 43 | 12.8
30101W | 43 | 12.8
30246 | 51 | 13.0
30286 | 32 | 12.6
30508 | 48 | 12.9
30589 | 51 | 13.0
30974 | 51 | 13.0
31000 | 28 | 12.5
31253 | 54 | 13.0
31412 | 36 | 12.7
31501 | 33 | 12.6
31867 | 28 | 12.4
32070 | 39 | 12.7
32237 | 29 | 12.5
32259 | 38 | 12.7
32347 | 54 | 13.0
33021A | 28 | 12.5
33334NE | 47 | 12.9
33334SW | 47 | 12.9
33636 | 29 | 12.5
33866 | 47 | 12.9
34445 | 45 | 12.9
34590 | 58 | 13.1
34745 | 37 | 12.7
35147 | 50 | 13.0
35769 | 57 | 13.1
36308 | 36 | 12.7
36443 | 38 | 12.7
36667 | 57 | 13.1
37271 | 48 | 12.9
37685 | 73 | 13.3
37773 | 38 | 12.7
38230A | 21 | 12.2
39570A | 54 | 13.0
39881A | 28 | 12.4
40040 | 68 | 13.2
40616 | 54 | 13.0
41708 | 45 | 12.9
42160 | 45 | 12.9
42618 | 23 | 12.3
43587A | 19 | 12.1
43947 | 28 | 12.4
44985 | 33 | 12.6
45391 | 26 | 12.4
45580 | 48 | 12.9
45759 | 50 | 13.0
46375 | 33 | 12.6
47127 | 27 | 12.4
48684 | 54 | 13.0
50060 | 87 | 13.4
51219 | 32 | 12.6
51295 | 42 | 12.8
52456 | 28 | 12.5
53505 | 58 | 13.1
53532 | 44 | 12.8
54046 | 46 | 12.9
54100 | 45 | 12.9
54351 | 44 | 12.8
54405 | 46 | 12.9
54718 | 46 | 12.9
55458A | 25 | 12.4
55918 | 38 | 12.7
56202 | 53 | 13.0
56303 | 41 | 12.8
56513 | 35 | 12.7
58781 | 30 | 12.5
58971 | 43 | 12.8
59360 | 41 | 12.8
59374 | 50 | 13.0
60298 | 39 | 12.7
62346 | 51 | 13.0
63935 | 50 | 12.9
64090 | 28 | 12.5
64324 | 35 | 12.6
65629 | 32 | 12.6
66348A | 43 | 12.8
66485 | 44 | 12.8
66550 | 38 | 12.7
68168 | 34 | 12.6
68284 | 300 | 14.5
69056 | 38 | 12.7
70088 | 43 | 12.8
70571A | 92 | 13.5
70571B | 92 | 13.5
72946 | 23 | 12.3
73226 | 43 | 12.8
73668A | 36 | 12.7
74011 | 46 | 12.9
74156 | 65 | 13.2
74567 | 93 | 13.5
75302 | 30 | 12.5
76218 | 26 | 12.4
76261 | 46 | 12.9
76349 | 49 | 12.9
76752 | 40 | 12.8
76765 | 61 | 13.1
76780 | 34 | 12.6
77024A | 77 | 13.3
77278 | 31 | 12.5
77407 | 30 | 12.5
78317 | 48 | 12.9
78660 | 50 | 13.0
79498A | 49 | 12.9
79726 | 48 | 12.9
80408 | 39 | 12.7
80536 | 51 | 13.0
80870 | 45 | 12.9
81040 | 33 | 12.6
82140 | 59 | 13.1
82939 | 38 | 12.7
83408 | 58 | 13.1
84209 | 72 | 13.3
84749 | 47 | 12.9
85426 | 61 | 13.1
85689 | 45 | 12.9
86133A | 42 | 12.8
86133B | 42 | 12.8
86460 | 41 | 12.8
86794 | 52 | 13.0
87680 | 39 | 12.7
88371 | 62 | 13.1
88446 | 69 | 13.2
88725 | 36 | 12.7
89055 | 36 | 12.7
89307 | 31 | 12.5
89813 | 27 | 12.4
90164 | 54 | 13.0
90905 | 32 | 12.6
91148 | 37 | 12.7
91204 | 52 | 13.0
93215 | 47 | 12.9
94028 | 52 | 13.0
94292 | 77 | 13.3
94426 | 53 | 13.0
95177 | 49 | 12.9
95364 | 82 | 13.4
95366 | 59 | 13.1
95980 | 53 | 13.0
96094 | 60 | 13.1
96497 | 53 | 13.0
96574 | 50 | 13.0
96679 | 66 | 13.2
96937 | 31 | 12.5
98078 | 41 | 12.8
98697 | 43 | 12.8
99404 | 53 | 13.0
99419 | 45 | 12.9
99505 | 33 | 12.6
100180A | 23 | 12.3
100180B | 23 | 12.3
101242 | 34 | 12.6
101444 | 41 | 12.8
103111 | 78 | 13.3
103847 | 29 | 12.5
104243 | 36 | 12.7
104923 | 40 | 12.7
104956 | 51 | 13.0
105087 | 58 | 13.1
105844 | 43 | 12.8
106156 | 31 | 12.5
106210 | 34 | 12.6
106252 | 37 | 12.7
106366 | 66 | 13.2
106510 | 51 | 13.0
107146 | 29 | 12.5
107705A | 30 | 12.5
107705B | 30 | 12.5
108653 | 51 | 13.0
108874 | 69 | 13.2
109628A | 84 | 13.4
109628B | 84 | 13.4
111398A | 36 | 12.7
111470 | 56 | 13.1
112001 | 55 | 13.0
112257 | 42 | 12.8
112735 | 58 | 13.1
112756 | 53 | 13.0
112959A | 59 | 13.1
113319 | 31 | 12.5
114060A | 37 | 12.7
114060B | 37 | 12.7
114174 | 26 | 12.4
114606 | 61 | 13.1
115231 | 53 | 13.0
115273 | 49 | 12.9
115349 | 45 | 12.9
115382 | 91 | 13.5
115519 | 61 | 13.1
115755 | 34 | 12.6
115762A | 60 | 13.1
116442 | 16 | 12.0
116443 | 17 | 12.0
116497 | 49 | 12.9
117858 | 60 | 13.1
118659 | 53 | 13.0
119054 | 50 | 12.9
119056 | 61 | 13.1
119550 | 60 | 13.1
120553 | 44 | 12.8
121298NW | 75 | 13.3
121298SE | 75 | 13.3
121320 | 33 | 12.6
122518 | 55 | 13.0
122652 | 37 | 12.7
123033B | 44 | 12.8
124019 | 49 | 12.9
124677A | 36 | 12.7
125056 | 42 | 12.8
126246A | 36 | 12.7
126246B | 36 | 12.7
126512 | 47 | 12.9
126583 | 34 | 12.6
126961 | 41 | 12.8
127825 | 58 | 13.1
128219 | 68 | 13.2
129209 | 50 | 13.0
129413A | 40 | 12.8
129814 | 42 | 12.8
130268 | 68 | 13.2
131179 | 39 | 12.7
132973 | 75 | 13.3
133161 | 35 | 12.7
134066A | 32 | 12.6
134066B | 32 | 12.6
135101A | 28 | 12.5
135101B | 28 | 12.5
135792A | 43 | 12.8
136925 | 46 | 12.9
138246 | 62 | 13.1
138573 | 31 | 12.5
138919 | 41 | 12.8
139018 | 79 | 13.3
139324 | 53 | 13.0
139457 | 47 | 12.9
139839 | 65 | 13.2
140209 | 65 | 13.2
140233 | 78 | 13.3
140324 | 55 | 13.0
140514 | 83 | 13.4
140750 | 68 | 13.2
141272 | 21 | 12.2
141529 | 55 | 13.0
142093 | 31 | 12.5
142229 | 41 | 12.8
142637 | 66 | 13.2
143291 | 26 | 12.4
144873 | 47 | 12.9
145229 | 33 | 12.6
145729 | 45 | 12.9
146588 | 45 | 12.9
146644 | 61 | 13.1
147044 | 36 | 12.7
147528 | 51 | 13.0
147750 | 40 | 12.8
148530 | 47 | 12.9
148816 | 41 | 12.8
149028 | 48 | 12.9
149380 | 94 | 13.5
149890 | 39 | 12.7
150554A | 45 | 12.9
150828B | 63 | 13.2
150933A | 44 | 12.8
152264 | 64 | 13.2
153627 | 43 | 12.8
153701 | 37 | 12.7
154417 | 20 | 12.2
154656 | 42 | 12.8
154931 | 55 | 13.0
155060 | 36 | 12.7
155193 | 61 | 13.1
155358 | 43 | 12.8
155423 | 44 | 12.8
156146 | 76 | 13.3
156893 | 68 | 13.2
156968 | 55 | 13.0
157089 | 39 | 12.7
157637 | 41 | 12.8
158226A | 69 | 13.2
158331 | 51 | 13.0
158332 | 30 | 12.5
159909 | 37 | 12.7
160013 | 42 | 12.8
161728 | 63 | 13.2
161848 | 38 | 12.7
162209 | 53 | 13.0
163609A | 50 | 12.9
164595 | 29 | 12.5
165173 | 33 | 12.6
165476 | 45 | 12.9
165672 | 44 | 12.8
166301 | 35 | 12.6
167081A | 49 | 12.9
169359 | 60 | 13.1
169748A | 52 | 13.0
169822 | 27 | 12.4
169889 | 38 | 12.7
170294 | 58 | 13.1
170469 | 65 | 13.2
171009 | 70 | 13.2
171067 | 25 | 12.4
171620 | 52 | 13.0
172310 | 36 | 12.7
172649 | 47 | 12.9
172867 | 66 | 13.2
173548A | 52 | 13.0
173548B | 52 | 13.0
174457 | 55 | 13.0
174719 | 28 | 12.5
175726 | 27 | 12.4
177305 | 43 | 12.8
178911B | 49 | 12.9
180684 | 59 | 13.1
181047A | 50 | 13.0
182619 | 33 | 12.6
182758 | 66 | 13.2
183341 | 45 | 12.9
183970 | 50 | 13.0
184403 | 44 | 12.8
184592 | 42 | 12.8
186704A | 30 | 12.5
187123 | 48 | 12.9
187548 | 45 | 12.9
187882 | 76 | 13.3
187897 | 33 | 12.6
187923A | 28 | 12.4
188015 | 53 | 13.0
188510 | 39 | 12.7
189067 | 43 | 12.8
189087 | 25 | 12.4
190404 | 16 | 11.9
190516A | 44 | 12.8
190594 | 43 | 12.8
190605 | 44 | 12.8
190609 | 44 | 12.8
191672 | 69 | 13.2
192367 | 129 | 13.8
195019A | 37 | 12.7
195034 | 28 | 12.5
196201 | 38 | 12.7
196885A | 33 | 12.6
198089 | 39 | 12.7
198416 | 60 | 13.1
200466E | 44 | 12.8
200466W | 44 | 12.8
200565 | 64 | 13.2
200746 | 44 | 12.8
201219 | 36 | 12.7
201891 | 35 | 12.7
202072 | 49 | 12.9
202108 | 27 | 12.4
203030 | 41 | 12.8
204277 | 34 | 12.6
204712 | 61 | 13.1
205702 | 57 | 13.1
206332 | 50 | 13.0
206374 | 27 | 12.4
206387A | 55 | 13.0
206658 | 48 | 12.9
207740 | 49 | 12.9
209262A | 46 | 12.9
209458 | 47 | 12.9
209858 | 55 | 13.0
209875 | 51 | 13.0
210388 | 43 | 12.8
210460 | 56 | 13.0
210462A | 54 | 13.0
210483 | 49 | 12.9
210553 | 45 | 12.9
211476 | 31 | 12.5
211786 | 42 | 12.8
212291 | 32 | 12.6
212858 | 55 | 13.0
214059 | 80 | 13.4
214435 | 49 | 12.9
214560 | 55 | 13.0
215257 | 42 | 12.8
215274 | 45 | 12.9
216625 | 44 | 12.8
216631 | 43 | 12.8
217165 | 44 | 12.8
218133 | 38 | 12.7
218172 | 74 | 13.3
218261 | 28 | 12.5
219172 | 46 | 12.9
220008 | 87 | 13.4
220077 | 77 | 13.3
220255 | 52 | 13.0
220334B | 37 | 12.7
220773 | 48 | 12.9
221477 | 59 | 13.1
221822 | 39 | 12.7
221851 | 23 | 12.3
221876 | 75 | 13.3
222033 | 50 | 13.0
222941A | 46 | 12.9
223061 | 45 | 12.9
223238 | 47 | 12.9
224156 | 29 | 12.5
225261 | 26 | 12.4
|  |