# On the gravitational origin of the Pioneer Anomaly

Siutsou I. A<EMAIL_ADDRESS>ICRANet, Pescara, Italy
65122, p-le della Repubblica, 10  ICRA and Dip. di Fisica, Universita di Roma
“Sapienza”, Roma, Italy
00185, p-le A. Moro 5  Stepanov Institute of Physics, NASB, Minsk, Belarus
220072, Nezalezhnastsi Av., 68  Tomilchik L. M  Stepanov Institute of Physics,
NASB, Minsk, Belarus
220072, Nezalezhnastsi Av., 68, Tel.: +375-29-4040838

###### Abstract

From Doppler tracking data and data on circular motion of astronomical objects
we obtain a metric of the Pioneer Anomaly. The metric resolves the issue of
manifest absence of anomaly acceleration in orbits of the outer planets and
extra-Pluto objects of the Solar system. However, it turns out that the
energy-momentum tensor of matter, which generates such a gravitational field
in GR, violates energy dominance conditions. At the same time the equation of
state derived from the energy-momentum tensor is that of dark energy with
$w=-1/3$. So the model proposed must be carefully studied by ”Grand-Fit”
investigations.

###### pacs:

04.25.Nx and 04.80.Cc and 04.50.Kd

##  I Introduction

Spacecrafts Pioneer 10 and 11 were launched in the early 1970’s for the
exploration of outer planets of the Solar system (see the special issue of
Science 183, No. 4122, 25 January 1974, especially Soberman1974 ; Anderson1974
). After the encounters with Jupiter and Saturn they followed hyperbolic
trajectories on leaving the Solar system. Because of rotational spin-
stabilization of these spacecrafts, which reduces the need for manoeuvres,
they represent unique experiments for testing our understanding of celestial
mechanics. The accuracy of acceleration measurements for the Pioneer
spacecrafts is about $10^{-10}$ m/s2 PioneerMissionPlan ; Scherer1997 .

During the flight spacecrafts were continuously tracked by Doppler effect on
retransmitted radio signals. Then data were fitted to theoretical ones
obtained in PPN approximation initially by ODP program of JPL (JPL’s Export
Planetary Ephemeris DE405 was used for planet motion).

But surprisingly above 10 a.u. of heliocentric distance the systematical
deviation of experimental and theoretical data was found Pioneer1998 . This
deviation can be described simply as a constant acceleration towards the Sun
with magnitude of about $8\cdot 10^{-10}$ m/s2. This value is the same —
within error limits — for all the spaceships Pioneer 10 and 11, Galileo and
Ulysses and for all distances from the Sun Pioneer2001 .

This coincidence has been interpreted as a hint of the gravitational — metric
— origin of the acceleration. But at the same time there are no signatures of
such an acceleration in the orbits of outer planets and other objects in the
Solar system. Inclusion of such acceleration leads to unavoidable deviations
from the observed planet positions Pioneer2006 ; Iorio2007 .

Many attempts to explain the anomaly were made during last 10 years. Some of
the recent work on this subject includes analyses of: the thermal radiation of
the Pioneers ISI:000263816800017 ; ISI:000261214100012 , the gravitational
attraction by the Kuiper Belt ISI:000238999400005 ; ISI:000239223200009 ;
ISI:000232936700007 , the cosmological origin of the Anomaly
ISI:000255093800006 ; ISI:000242327900022 ; ISI:000255524300012 ;
ISI:000258636700088 ; ISI:000236229700005 ; ISI:000246464000017 , the
influence of multipole moments of the Sun 2005AIPC..758..129Q , the clocks
acceleration ISI:000227551100006  and many proposes of modified gravity
ISI:000238120300014 ; ISI:000254557000003 ; ISI:000259935000020 ;
ISI:000257329300072  including even laboratory investigations on very small
acceleration dynamics ISI:000245691400011  and interesting endeavors to
constrain some parameters of modified gravity theories by the known value of
the Pioneer Anomaly ISI:000248810500006 ; ISI:000262356900001 .

In the frame of metric theories of gravitation there is an attractive
possibility to explain the Pioneer Anomaly by metric perturbation, preserving
at the same time the character of planet motion. It is possible because the
Pioneers’ trajectories are very different from planet orbits: spacecrafts
leave the Solar system almost in a radial direction, while planets orbit the
Sun almost circularly. The potential possibility of such an approach was noted
independently in a series of manuscripts 2005CQGra..22.2135J ;
2006CQGra..23..777J ; 2006CQGra..23.7561J , but authors of these works didn’t
analyze the origin of the metric. Our approach is closer to that of Kjell
Tangen Tangen2007 , but with more rigor because we don’t neglect perturbation
of the space components of metric.

The goal of paper is to find the static space-time metric close to
Schwarzschild one, in which radial motion of test bodies shows the Pioneer
Anomaly, but circular motion doesn’t. For this purpose in section II we
develop and discuss an algorithm of metric determination from data on radial
and circular motions. Metric determination does not make use of Einstein
equations so it is applicable to any pure metric theory of gravity in
terminology of Will et. al. Will1993 ; Will2006 . Then we apply the method in
the case of the Pioneer Anomaly, starting from the Schwarzschild space-time
and recover the properties of matter forming such a metric within GR (section
III). Finally some concluding remarks are made.

##  II Space-time determination from radial and circular motions

###  II.1 Metric choice and time definition

We begin with the interval with 3 metric functions $\tau$, $\rho$ and $\sigma$
(for simplicity it was taken $c=1$)

$\displaystyle
ds^{2}=e^{\tau(r)}dt^{2}-e^{\rho(r)}dr^{2}-e^{\sigma(r)}r^{2}(d\theta^{2}+\cos^{2}\theta
d\varphi^{2}),$ (1)

and then find the gauge relation between them for maximal simplification. The
metric functions $\tau$, $\rho$ and $\sigma$ will be referred as time, radial
and transverse metric function or coefficient, respectively. We consider
radial and circular motions in such a space-time separately and find
connections between metric functions following from the known properties of
the motion. But first of all we must recall some time convention.

The metric (1) is written in the form that is consistent with global clock
synchronization, in fact $\partial/\partial t$ being timelike Killing vector.
So in the approach proposed when the cosmological effects is totally neglected
and Solar system is supposed to be placed in a space-time, that is Minkowskian
at spatial infinity, the coordinate time is the astronomical ephemerides time
ET (up to a multiplier). This is the usual approximation used for PPN-
ephemeride calculations.

There is a difficulty, because the perturbations required by the Pioneer
Anomaly grow with the radial distance, so the perturbed space-time is not
asymptotically flat. But it is not a big problem because before the metric
perturbations grow significantly the space-time has a wide nearly-flat region
in which we can choose almost Minkowskian observers and coordinates. So
further if we talk about ”spatial infinity”, we mean this wide region, in the
Solar system scale the effects of difference between this approximation and
rigorous treatment are negligible. The same problem with the same solution is
arising in the PPN-approximation then we must place the system not in
Minkowskian background but in the cosmological one. Additionally, as it can be
shown, in PPN-approach the cosmological effects, such as mutual acceleration
of geodetically moving bodies, have the second order in $H$. Therefore even
while the Pioneer Anomaly acceleration is nearly equal to $cH$, in the
framework of pure metric theories of gravitation there is no possibility to
link it to the cosmological expansion (early but almost exhaustive analysis of
the problem was made by R. C. Tolman (Tolman, , §§153–156), for the recent
work on subject see the articles mentioned in the Introduction and
additionally FahrSiewert2006 ; MizonyLachiezeRey2005 ; Licht2001  and
references within).

The radial coordinate rescaling $r\rightarrow f(r)$ changes all metric
functions, giving a possibility to imply gauge conditions on the metric
coefficients. But there are two invariants, i.e. physically measurable
quantities, which characterize the distance from a given point to the center
of space symmetry. Firstly it is an atomic time rate in comparison with the
coordinate time rate (or atomic time rate on ”spatial infinity”) $e^{\tau/2}$,
and secondly it is an area of a sphere of points equidistant to the center of
the space $4\pi r^{2}e^{\sigma}$. So the numerical values of time and
transverse metric coefficients have clear physical meaning for a given space-
time point and only $\rho(r)$ can take arbitrary values. Usual choices include
$\rho\equiv\sigma$ corresponding to isotropic coordinates of PPN-approximation
and implicit on $\rho(r)$ relation $\sigma\equiv 0$ corresponding to
Schwarzschild coordinates.

###  II.2 Radial motion and its description by Doppler radio tracking

Radial motion in the space-time is fully described by the energy
$g_{tt}\frac{dt}{ds}=e^{\tau(r)}u^{0}=k=const$ and the 4-velocity length
conservation $e^{\tau(r)}u^{0^{2}}-e^{\rho(r)}u^{1^{2}}=\varepsilon,\
\varepsilon=0$ or 1 for electromagnetic waves and test bodies, respectively:

$\displaystyle\frac{dt}{dr}=\frac{e^{\frac{\rho(r)-\tau(r)}{2}}}{\sqrt{1-\varepsilon
e^{\tau(r)}/k^{2}}},$ (2)

the constant $k$ being connected with the velocity $v$ of the spacecraft on
”space infinity”

$\displaystyle k^{2}=\frac{1}{1-v^{2}}.$ (3)

The relation (2) can be integrated to give us $t(r)$ dependence which in turn
can be inverted giving $r(t)$. But there is a gauge freedom in the result
because we can choose $\rho(r)$ freely. Moreover, this relation essentially
involve both physical $\tau$ and unphysical $\rho$.

Our goal is to determine the metric functions from observations. Direct
results of experimentation in astronomy and cosmology are the measurements of
externally originated signals received on the world line of an observer. These
signals have mostly electromagnetic character and can originate from some
external source (e. g. the light emitted by the Sun and reflected by a planet)
or be emitted by the observer him/herself and then returned to him/her after
some interaction with outer bodies (as in case of radar measurements). The
latter case is considered here as corresponding to the real situation
Pioneer2001 .

The scheme of Doppler tracking used in the Pioneer experiments is very simple
conceptually: an electromagnetic signal, emitted from the world line of the
observer, is reflected back by the mirror on the world line of the spacecraft
and then compared with the initial one again on the observer world line. To be
more precise, the monochromatic electromagnetic signal, obtained from the high
precision hydrogen maser (Pioneer2001, , subsection III.A), is emitted by the
antennae on Earth in the direction of the spacecraft. This signal is detected
by the spacecraft, amplified and reemitted back to Earth, where the measured
waveform of arrived signal is compared with the emitted one to obtain the red
shift of the signal and the time of signal travel (the very detailed
description of the process can be found in Pioneer2001 ).

Now we must describe the Doppler tracking and the signal time arrival analysis
in this space-time. In the geometric optics approximation (which is applicable
for the case considered) the Doppler shift is governed simply by the ratio
between scalar products of the 4-velocity on world lines and the null wave
vector of the signal, parallel transported along the null geodesic line
between emitter and receiver:

$\displaystyle\frac{\nu_{r}}{\nu_{e}}=\frac{s_{e}}{s_{r}}=\frac{\vec{\mathstrut
u}_{r}\cdot\vec{k}_{r}}{\vec{\mathstrut u}_{e}\cdot\vec{k}_{e}},$ (4)

where $\nu_{r}$ and $\nu_{e}$ are received and emitted frequencies, measured
by the standard atomic clocks,
$s_{r}$ and $s_{e}$ are proper times of one cycle of oscillation,
$u_{r}$ and $u_{e}$ are 4-velocities of receiver and emitter,
$k_{r}$ and $k_{e}$ are tangential null vectors (wave vector), parallel
transported along the path of the signal.

Atomic clock time deviations from ephemerides time along with all known
effects of Earth motion were taken into account during the data processing
(see Pioneer2001 ), so for the description of such a small deviation like the
Pioneer Anomaly it is sufficient to use the simple model, in which the emitter
of the initial signal and the receiver of the retranslated signal are fixed at
constant distances from the Sun on the line from the Sun to apparatus. The
signal is emitted from this ”fixed” Earth at $r_{0}$ and $t-t_{p}$, received
by the spaceship at $r$ and $t$, amplified, exactly retransmitted back to
Earth and finally compared with the initial frequency on the ”fixed” Earth
again at $r_{0}$ and $t+t_{p}$ ($t_{p}$ is the time of signal propagation, the
same for forward and backward directions).

As it can be shown easily, the frequency $\nu_{r}$ received on Earth is
connected to the initially emitted $\nu_{e}$ as

$\displaystyle\nu_{r}=\nu_{e}\frac{1-\sqrt{1-e^{\tau(r)}/k^{2}}}{1+\sqrt{1-e^{\tau(r)}/k^{2}}}.$
(5)

This expression can be readily reduced to special relativistic one in the case
of $e^{\tau}\equiv 1$.

As a relation between physical quantities only, this equation does not involve
arbitrary unphysical $\rho$ function. The problem of gauge choice comes with
the definition of $r(t)$: from (5) one can determine $e^{\tau(r(t))}$, but
because the radial coordinate $r$ (and consequently $r(t)$) is arbitrary, the
radial dependence of the time metric coefficient remains gauge dependent. It
is interesting also that this relation cannot be represented by a power series
in terms of small deviations of $e^{\tau(r)}$ and $k$ from 1. The transverse
space metric coefficient $\sigma(r)$ naturally cannot be determined from the
radial motion only.

So we come to an unavoidable alternative of a-priory $r(t)$ definition or
a-priory imposing some gauge condition on $\rho(r)$. In each case the
remaining function is defined by the experimental data, and our goal now is to
find in which case the process of metric restoration can be done without
unnecessary complications. In the next subsection we consider these
possibilities in some details and then show that the best choice is the latter
case, i.e. imposing a gauge.

###  II.3 General formulae and coordinate choice

While the time of signal arrival to the spaceship $t$ can be easily determined
as a half-sum of observed times of emitting $t_{e}$ and receiving $t_{r}$ of
the signal at the ”fixed” Earth

$\displaystyle t=\frac{t_{r}+t_{e}}{2},$ (6)

the corresponding $r$ determination is not so trivial task. In general we can
measure only the time of signal travel from the ”fixed” Earth to the spaceship
as a half-difference between observed times of sending and receiving of the
signal

$\displaystyle
t_{p}=\int_{r_{0}}^{r}{e^{\frac{\rho(r)-\tau(r)}{2}}}\,dr=\frac{t_{r}-t_{e}}{2}.$
(7)

These are results of a different method of tracking — signal time arrival
analysis, which in essence represents integration of the Doppler data.

So to recover $\rho(r)$ from a given $r(t)$ we must firstly find $\tau(r)$ by
(5) from the observed redshift, and then solve an integral equation above. On
the other hand, to find $r(t)$ from a given $\rho(r)$ it is sufficient to
solve a non-integral equation following from relation (2) for the spacecraft
motion

$\displaystyle\int_{t_{0}}^{t}e^{\frac{\tau(r(t))}{2}}\sqrt{1-e^{\tau(r(t))}/k^{2}}\,dt=\int_{r_{0}}^{r}{e^{\frac{\rho(r)}{2}}}\,dr,$
(8)

where $t_{0}$ is the time when the spacecraft leaves ”fixed” Earth. The left-
hand side of the relation can be found totally from the observed frequency
shifts by (5), and the right-hand side is a known function of $r$ with given
$\rho(r)$.

Consequently maximal simplification of the problem is reached in the case of
a-priory given radial metric function $\rho(r)$. It is nonsense to define it
dependent on still unknown time and transverse metric functions, with one
interesting exclusion: if $\tau(r)\equiv\rho(r)$ then $r(t)$ can be recovered
from (7) simply as

$\displaystyle r=r_{0}+\frac{t_{r}-t_{e}}{2}.$ (9)

This choice of coordinates known as light or null coordinates is not so usual
as Schwarzschild ($\sigma\equiv 0$) or isotropic ($\rho\equiv\sigma$)
coordinates but it is the most suitable one for the situation. Both
abovementioned choices are especially inappropriate here because they rely on
transversal metric function that does not reveal itself in pure radial
motions.

###  II.4 Circular motion

We know that the near-circular motion of outer Solar system objects (i.e.
Neptune or Pluto) is unperturbed by the Pioneer Anomaly acceleration
Pioneer2006 ; Iorio2007 . This gives us a way to determine the transversal
metric coefficient and therefore to find metric completely.

The angular velocity of circular motion ($\theta=0,\ \phi=\omega t$) in the
considered space-time is defined by the ratio of derivatives of time and
transversal space metric coefficients

$\displaystyle\omega^{2}(r)=\frac{(e^{\tau(r)})^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}}=\frac{(e^{\tau(r(t))}){}\dot{}}{(r^{2}e^{\sigma(r(t))}){}\dot{}},$
(10)

where as usual prime denotes differentiation with respect to radial
coordinate, and dot denotes differentiation in time. So there is no dependence
on the radial metric part at all. Moreover as $\omega(r(t))$ and
$(e^{\tau(r(t))}){}\dot{}$ is directly observable, so the transverse metric
coefficient $r^{2}e^{\sigma(r(t))}$ can be obtained by a simple integration
without any notion of the radial metric function.

So again recalling the radial motion equation (2) we conclude that all gauge
conditions for $\rho(r)$ involving transverse metric coefficient are not
convenient for treatment of the Pioneer Anomaly, because all such conditions
lead to coupling of equations (5) and (10) which can be solved independently
otherwise.

###  II.5 Final list of relations and concluding remarks

So we work in light or null coordinates $\tau(r)\equiv\rho(r)$, then:

$\displaystyle\frac{dt}{dr}=\frac{1}{\sqrt{1-\varepsilon e^{\tau(r)}/k^{2}}}.$
(11) The trajectory of the spacecraft $r(t)$ is recovered simply from time of
sending $t_{e}$ and arrival $t_{r}$ of signal $\displaystyle
t_{p}=r-r_{0}\quad\Rightarrow\quad t=\frac{t_{r}+t_{e}}{2},\qquad
r=r_{0}+\frac{t_{r}-t_{e}}{2},$ (12)

and the observed redshift of the signal

$\displaystyle
z(t)=\frac{\nu_{e}-\nu_{r}}{\nu_{e}}=\frac{\Delta\nu}{\nu_{e}}=\frac{2}{(1-e^{\tau(r(t))}/k^{2})^{-1/2}+1},$
(13)

can be immediately transformed into time metric coefficient

$\displaystyle
e^{\tau(r(t))}=k^{2}\left[1-\left(\frac{z(t)}{2-z(t)}\right)^{2}\right]=4k^{2}\frac{(1-z)}{(2-z)^{2}}.$
(14)

The transverse space metric coefficient is defined by the dependence of
angular velocity $\omega$ on radial coordinate $r$

$\displaystyle
r^{2}e^{\sigma(r)}=r_{0}^{2}e^{\sigma(r_{0})}-\int_{r_{0}}^{r}\frac{4k^{2}z(r)z^{\prime}(r)}{(2-z(r))^{3}\omega^{2}(r)}dr,$
(15)

which for the space-time of the Pioneer Anomaly should be the same as in the
Schwarzshild field (or very close to it). So basing on the Schwarzshild metric
one can find such perturbations of metric coefficients, that the circular
motion remains unperturbed, but the radial one shows small deviation — the
Pioneer Anomaly.

It is worth noting that the sentence ”the circular motion remains unperturbed”
denotes exactly the following: if, neglecting all mutual planet disturbances,
the period of circularly orbiting planet will be measured in the units of
ephemerides time and the distance of planet from the baricenter of the Solar
system will be found by analyzing light propagation times on straight lines
from the Sun (null coordinates!) then the values of the period and the
distance will exactly be the same as needed for the 3rd Kepler law to hold —
exactly as in the Schwarzshild field.

##  III The Pioneer Anomaly and its source in GR

###  III.1 Schwarzschild space-time in light coordinates

Radius $r$ in light coordinates of the Schwarzschild field is related to the
usual Schwarzschild radial coordinate $r_{s}$ as

$\displaystyle r=r_{s}+r_{g}\ln\left(\frac{r_{s}}{r_{g}}-1\right),\quad
r_{s}=r_{g}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right).$ (16)

The interval in null coordinates is as follows

$\displaystyle\begin{split}ds^{2}=&\frac{\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}{1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}(dt^{2}-dr^{2})\\\
&-r_{g}^{2}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{2}(d\theta^{2}+\cos^{2}\theta
d\varphi^{2}),\end{split}$ (17) so that $\displaystyle
e^{\tau(r)}=e^{\rho(r)}=\frac{\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}{1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)}=r_{g}\mathop{W}\nolimits^{\prime}_{r}\left(e^{\frac{r}{r_{g}}-1}\right),$
(18) $\displaystyle
e^{\sigma(r)}=\frac{r_{g}^{2}}{r^{2}}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{2},$
(19) where $W(x)$ is the so called multiplicative logarithm or Lambert $W$
function $\displaystyle W(x)e^{W(x)}=x.$ (20)

###  III.2 The Pioneer Anomaly. Radial perturbation

The Pioneer Anomaly is the linear in ephemerides time ET deviation of the
experimentally obtained frequency of received signal $\nu_{r}$ from the
”modelled” one $\nu_{m}$

$\frac{d}{d\text{{ET}}}(\nu_{r}-\nu_{m})=-\nu_{e}\frac{2a_{P}}{c},$ (21)

where $a_{P}\sim 8\cdot 10^{-10}\ \mbox{m/s}^{2}$ is the ”unmodelled”
acceleration Pioneer2001 . The ephemerides time coincides with time coordinate
$t$ of the metric considered (as described earlier). For the most part of
range of the Pioneer Anomaly found (from $\sim 15$ to $\sim 80$ a.u.) the
deviation of the Pioneers’ orbits from pure radial motion is comparable to or
even below experimental uncertainty in the acceleration: it can be estimated
roughly as a ratio of the semi-major axis $a$ absolute value to heliocentric
distance $r$

$\frac{|a|}{r}\lesssim\frac{10^{9}\text{km}}{40\cdot 150\cdot
10^{6}\text{km}}\simeq 17\%,$ (22)

while experimental error in the acceleration is $1.33/8.74\simeq 15\%$ (see
Appendix of Pioneer2001 ). It should be noted that this uncertainty prevents
Anderson et. al. from determining the direction of the acceleration: to Earth
or to the Sun (see beginning of section VII and especially note 73 of
Pioneer2001 ).

The modelled frequency and velocity of the spacecraft are

$\displaystyle\nu_{m}=\nu_{0}\;\frac{1+\frac{1}{W(e^{\frac{r}{r_{g}}-1})}}{1-v^{2}}\left(1-\sqrt{1-\frac{1-v^{2}}{1+\frac{1}{W(e^{\frac{r}{r_{g}}-1})}}}\right)^{2},$
(23)
$\displaystyle\left(\frac{dr}{dt}\right)_{m}=v_{m}(r)=\sqrt{\frac{v^{2}+1/W(e^{\frac{r}{r_{g}}-1})}{1+1/W(e^{\frac{r}{r_{g}}-1})}},$
(24)

so accurately expanding the expression (14) for the time metric coefficient
from the red shift $z(r(t))$, one can find that given the accuracy of the
Pioneer Anomaly measurements one can simply use the relation

$\displaystyle\delta e^{\tau(r)}\simeq-\frac{z(r)\;\delta
z(r)}{2}=a_{P}\;z(r)\Delta t(r),$ (25)

where $\Delta t=t(r)-t_{0}$ is the time from the start of the Pioneer Anomaly
(we suppose that for $r<r_{0}$ the metric coincides with the Schwarzschild
one, so before $t_{0}=t(r_{0})$ there is no anomalous acceleration), $\delta
z(r)$ is a deviation of observed red shift from the modelled one.

In the first approximation $t(r)$ dependence can be replaced with the
”modelled” time, which to the accuracy of the measurements is the same as in
the Newtonian case

$t_{m}(r)=t(r_{0})+\int_{r_{0}}^{r}\frac{dr}{\dot{r}}\simeq\\\ \simeq
t_{0}+\frac{r}{v^{2}}\sqrt{v^{2}+\frac{r_{g}}{r}}-\frac{r_{g}}{v^{3}}\sinh^{-1}\left(\sqrt{\frac{r}{r_{g}}}v\right).$
(26)

Finally inserting this and modelled $z(r)$ into the equation (25) we arrive to
the perturbation of the time metric coefficient

$\delta e^{\tau(r)}=2a_{P}\Biggl{(}r-C\sqrt{v^{2}+\frac{r_{g}}{r}}+\\\
+\frac{r_{g}}{v^{2}}\left[1-\sqrt{1+\frac{r_{g}}{r\;v^{2}}}\sinh^{-1}\left(\sqrt{\frac{r}{r_{g}}}v\right)\right]\Biggr{)},$
(27)

where $C$ is a constant which can be determined from $r_{0}$ and $v$. The
perturbation appears to be non-linear in $r$, but for the Pioneer 10/11
parameters the deviation from linearity is buried deep in the experimental
errors. It is illustrated by the figure 1, which shows the deviation of the
time metric coefficient compared to the ”naive” post-Newtonian approach, where
one simply adds to the gravitational potential $\Phi(r)$ a term linear in
radius and use $e^{\tau(r)}\simeq 1+\Phi(r)$. As we can see, the difference is
mainly in the slope, all the graphs are nearly linear. Moreover, the relative
value of the deviation from linearity is decreasing with radial distance. So
the linear approximation $\delta e^{\tau(r)}\simeq 2\eta a_{P}(r-r_{0})$ is
sufficient for the Pioneer Anomaly explanation. The only difference between
this more accurate result and the ”naive” post-Newtonian one is the presence
of $\eta$, which is always less then 1 (see table 1).

Figure 1: Metric perturbation $\delta e^{\tau(r)}$ of the Pioneer Anomaly for $v$ from 5 km/s to 50 km/s in 5 km/s steps (solid lines from bottom to top) for the metric matching the Schwarzschild metric at 12 a. u., compared to the ”naive” post-Newtonian one (dashed line) Table 1: The quantity $1-\eta$ of the best linear approximations $\delta e^{\tau(r)}\simeq 2\eta a_{P}(r-r_{0})$ on the interval $r_{0}\leq r\leq 70$ a. u. for different velocities $v$ and metric matching distances $r_{0}{}^{*}$ $r_{0}$, a. u. |  $v$, km/s
---|---
| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50
10 | 0.175 | 0.098 | 0.057 | 0.036 | 0.025 | 0.018 | 0.013 | 0.010 | 0.008 | 0.007
15 | 0.146 | 0.077 | 0.044 | 0.027 | 0.018 | 0.013 | 0.010 | 0.008 | 0.006 | 0.005
20 | 0.122 | 0.062 | 0.034 | 0.021 | 0.014 | 0.010 | 0.007 | 0.006 | 0.005 | 0.004
∗ $r_{0}$ is the radial coordinate at which the metric coefficients coincides
with the Schwarzschild ones.

###  III.3 Transversal perturbation leaving planet orbits unchanged. Necessity
of the central source

Now we assume that in the perturbed space-time planets orbit with the same
periods as in the unperturbed Schwarzschild solution, so that there is no
signatures of the Pioneer acceleration in the orbits of planets. So the
angular velocity dependence $\omega(r)$ must be the same as in the
Schwarzschild case (see, e. g., eq. (25.40) of MTW )

$\displaystyle\omega_{s}^{2}(r)=\frac{{e^{\tau(r)}}^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}}=\frac{1}{2r_{g}^{2}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{3}}=\frac{r_{g}}{2r_{s}^{3}},$
(28)

and the perturbation in $e^{\tau}$ must lead to such a perturbation in
$r^{2}e^{\sigma}$ that $\omega^{2}(r)$ remains invariant. In the perturbed
space-time by the general formula (10) we have from simple mathematics

$\omega_{s}^{2}(r)=\frac{{e^{\tau(r)}}^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}}=\omega^{2}(r)=\frac{(e^{\tau(r)}+\delta
e^{\tau(r)})^{\prime}}{(r^{2}e^{\sigma(r)}+\delta(r^{2}e^{\sigma(r)}))^{\prime}}=\\\
=\frac{(e^{\tau(r)})^{\prime}+(\delta
e^{\tau(r)})^{\prime}}{(r^{2}e^{\sigma(r)})^{\prime}+(\delta(r^{2}e^{\sigma(r)}))^{\prime}}=\frac{(\delta
e^{\tau(r)})^{\prime}}{(\delta(r^{2}e^{\sigma(r)}))^{\prime}}.$ (29)

So the perturbation of the transverse metric coefficient is

$\displaystyle\delta(r^{2}e^{\sigma(r)})=r^{2}\delta
e^{\sigma(r)}=\int_{r_{0}}^{r}\omega^{-2}(r)\delta({e^{\tau(r)}})^{\prime}\,dr,$
(30)

and finally

$\delta
e^{\sigma(r)}=\frac{4a_{P}\eta\,r_{g}^{2}}{r^{2}}\int_{r_{0}}^{r}\left(1+\mathop{W}\left(e^{\frac{r}{r_{g}}-1}\right)\right)^{3}\,dr\simeq\\\
\simeq\frac{4a_{P}\eta\,r_{g}^{2}}{r^{2}}\int_{r_{0}}^{r}\left(\frac{r}{r_{g}}\right)^{3}dr=\frac{4a_{P}\eta(r^{4}-r_{0}^{4})}{r^{2}r_{g}}.$
(31)

In the first approximation the perturbation of $r^{2}e^{\sigma(r)}$ grows
quartically in radius.

Note the gravitational radius of the source $r_{g}$ in the answer. So _the
effect of the Pioneer Anomaly can be reproduced only by perturbations of
Schwarzschild space-time and not Minkowski one_. So the gravitational
explanation of the Pioneer Anomaly can be obtained without the equivalence
principle violation required by various authors Pioneer2006 ; Iorio2007 ;
Tangen2007 .

###  III.4 Matter corresponding to the obtained metric in General Relativity

One can try to find the gravitational field theory, that gives equations of
the gravitational field allowing the solution found for the weak field of a
point mass. But we think that it is not very promising because there are no
reliable experimental evidence in favor of any gravitational theory other than
General Relativity.

Instead we find the properties of matter surrounding the point mass in GR that
can generate the obtained metric. Because in the scale of Solar system
experiments the influence of cosmological constant is negligible, for the
determination of matter one can use the Einstein equations of the form

$G_{ij}=R_{ij}-\frac{1}{2}Rg_{ij}=\kappa T_{ij},\qquad\kappa=\frac{8\pi
G}{c^{4}}.$ (32)

Using for simplicity the metric in the form

$ds^{2}=e^{\tau(r)}dt^{2}-e^{\rho(r)}dr^{2}-e^{\sigma(r)}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (33)

one arrives at the Einstein tensor

$\displaystyle G_{ij}=\frac{e^{-\rho}}{4}\Bigl{(}\lambda_{t}T_{i}\otimes
T_{j}-\lambda_{s}S_{i}\otimes S_{j}-\lambda g_{ij}\Bigr{)},$ (34)
$\displaystyle S_{i}=\Bigl{\\{}0,e^{\frac{\rho}{2}},0,0\Bigr{\\}},\qquad
T_{i}=\Bigl{\\{}e^{\frac{\tau}{2}},0,0,0\Bigr{\\}},$ (35) $\displaystyle-
S_{i}S^{i}=T_{i}T^{i}=1,\qquad S^{i}T_{i}=0,$ (36)
$\displaystyle\lambda_{t}=4e^{\rho-\sigma}+\left(\rho^{\prime}-2\sigma^{\prime}-\tau^{\prime}\right)\left(\sigma^{\prime}-\tau^{\prime}\right)+2\left(\tau^{\prime\prime}-\sigma^{\prime\prime}\right),$
(37)
$\displaystyle\lambda_{s}=4e^{\rho-\sigma}-\tau^{\prime}\left(\sigma^{\prime}-\tau^{\prime}\right)-\rho^{\prime}\left(\sigma^{\prime}+\tau^{\prime}\right)+2\left(\tau^{\prime\prime}+\sigma^{\prime\prime}\right),$
(38) $\displaystyle\lambda=\sigma^{\prime
2}+\sigma^{\prime}\tau^{\prime}+\tau^{\prime
2}-\rho^{\prime}\left(\sigma^{\prime}+\tau^{\prime}\right)+2\left(\tau^{\prime\prime}+\sigma^{\prime\prime}\right).$
(39)

Expanding to the first power of $a_{P}$ one obtains

$\displaystyle\lambda_{t}=-96\,\frac{a_{P}\eta}{r_{g}},\quad\lambda_{s}=-32\,\frac{a_{P}\eta}{r_{g}}\,\frac{r_{0}^{4}}{r^{4}},$
$\displaystyle\lambda=16\,\frac{a_{P}\eta}{r_{g}}\left(3-\frac{r_{0}^{4}}{r^{4}}\right).$
(40)

The algebraic type of the energy-momentum tensor at spatial infinity is that
of an ideal fluid (by $\lambda_{s}\rightarrow 0$) with constant positive
pressure

$\displaystyle p=\frac{e^{-\tau}}{4\kappa}\lambda\rightarrow
12\,\frac{a_{P}\eta}{\kappa r_{g}}>0,$ (41) but _negative energy density_
$\displaystyle{\rho=\frac{e^{-\tau}}{4\kappa}(\lambda_{t}-\lambda)\rightarrow-36\,\frac{a_{P}\eta}{\kappa
r_{g}}<0.}$ (42)

It is worth noting that relation between $p$ and $\rho$ is as for an
ultrarelativistic fluid except for the sign: instead of $p=\rho/3$ one has
asymptotically $p=-\rho/3$. This is a typical equation of state of dark energy
with parameter $w=-1/3$. It is interesting that such a fluid does not change
the cosmological dynamics of Friedman-Lemaître universe (see, e.g.,
(PeeblesRatra2003, , III.E)).

##  IV Conclusions

In this work we show that it is possible to perturb time metric coefficient of
the Schwarzschild space-time in such a way that the Pioneer Anomaly is
reproduced. Moreover, because planet motion is governed also by anther
component of the metric, we can tune it so that circular orbits is not
disturbed. This result applies in any pure metric theory of gravitation where
test bodies follow geodetics of the metric. It is deduced that the
perturbation of the time metric coefficient can be taken linear in $r$ without
contradiction with the accuracy of experimental data obtained up to now.

Assuming the validity of the General Relativity, we find out the energy-
momentum tensor generating the metric obtained. The tensor corresponds to an
ideal fluid, however having negative energy density. It is interesting to note
that an exact static solution with spherical symmetry is known for the ”fluid”
with $3p+\rho=0$ (StanjukovichMel'nikov1983, , §8.5). This ”fluid” does not
interact with the ordinary matter besides its gravitational influence on the
metric, so it much like WIMPs or scalar field of gravitational theories of
Brans-Dicke type. Thus the ordinary matter including spacecrafts and planets
is moving geodesically.

Naturally the found ”fluid” does change the planet orbits (if they are not
strictly circular) and light rays paths. The model proposed must be carefully
studied in view of the ”Grand-Fit” investigations Pitjeva2005 ;
2008AIPC..977..254S , but direct measurements from the planned missions for
testing General Relativity in space are preferable 2007arXiv0711.0304W ;
2005gr.qc…..6104L ; 2005gr.qc…..6139T . The absolute value of effects for the
perturbation found as well as for exact solution of Stanjukovich and Ivanov
will be studied in the forthcoming paper.

The analysis presented in this paper can encourage someone to find out which
of the known alternatives to GR can reproduce the metric found or to invent
some new theory which can do it. It is possible, but in our opinion the
Pioneer Anomaly has some simple explanation by conventional and non-
gravitational physics, which is not found yet. So the question of deep
theoretical grounds for the existence of the ”fluid” in GR or of development
of some new theory of gravitation based on the metric obtained is not in the
scope of our article. Instead we point out that negative energy density of the
”fluid” is in a direct contradiction with the properties of conventional
matter. One interesting possibility for the ”fluid” is dark energy, which has
the right equation of state. Therefore we suppose that at present the metric
(gravitational) origin of the Pioneer Anomaly cannot be ruled out.

###### Acknowledgements.

The authors thank participants of the Theoretical Physics Laboratory seminar
for helpful discussions and fruitful remarks, and also acknowledge the creator
of the RGTC package — S. Bonanos. One of us (SIA) thanks G. Vereshchagin for
the help in preparation of the present version of manuscript.

## References

  * (1) R.K. Soberman, S.L. Neste, K. Lichtenfeld, Science 183(4122), 320 (1974)
  * (2) J.D. Anderson, G.W. Null, S.K. Wong, Science 183(4122), 322 (1974)
  * (3) _Pioneer Extended Mission Plan, Revised, NASA/ARC document No. PC-1001_ (NASA, Washington, D.C., 1994)
  * (4) K. Scherer, H. Fichtner, J.D. Anderson, E.L. Lau, Science 278, 1919 (1997)
  * (5) J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto, S.G. Turyshev, Physical Review Letters 81(14), 2858 (1998).  DOI 10.1103/PhysRevLett.81.2858
  * (6) J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto, S.G. Turyshev, Physical Review D 65(8), 082004 (2002).  DOI 10.1103/PhysRevD.65.082004
  * (7) L. Iorio, G. Giudice, New Astronomy 11(8), 600 (2006).  DOI 10.1016/j.newast.2006.04.001
  * (8) L. Iorio, Foundations of Physics 37(6), 897 918 (2007)
  * (9) V.T. Toth, S.G. Turyshev, Physical Review D 79(4), 043011 (2009).  DOI 10.1103/PhysRevD.79.043011
  * (10) O. Bertolami, F. Francisco, P.J.S. Gil, J. Paramos, Physical Review D 78(10), 103001 (2008).  DOI 10.1103/PhysRevD.78.103001
  * (11) O. Bertolami, P. Vieira, Classical and Quantum Gravity 23(14), 4625 (2006).  DOI 10.1088/0264-9381/23/14/005
  * (12) J.A. De Diego, D. Nunez, J. Zavala, International Journal of Modern Physics D 15(4), 533 (2006).  ArXiv:astro-ph/0503368
  * (13) M. Nieto, Physical Review D 72(8), 083004 (2005).  DOI 10.1103/PhysRevD.72.083004
  * (14) H.J. Fahr, M. Siewert, Naturwissenschaften 95(5), 413 (2008).  DOI 10.1007/s00114-007-0340-1
  * (15) M. Carrera, D. Giulini, Classical and Quantum Gravity 23(24), 7483 (2006).  DOI 10.1088/0264-9381/23/24/019
  * (16) E. Hackmann, C. Lämmerzahl, Physical Review Letters 100(17), 171101 (2008).  DOI 10.1103/PhysRevLett.100.171101
  * (17) E. Hackmann, C. Lämmerzahl, Physical Review D 78(2), 024035 (2008).  DOI 10.1103/PhysRevD.78.024035
  * (18) V. Kagramanova, J. Kunz, C. Lämmerzahl, Physics Letters B 634(5-6), 465 (2006).  DOI 10.1016/j.physletb.2006.01.069
  * (19) M. Lachieze-Rey, Classical and Quantum Gravity 24(10), 2735 (2007).  DOI 10.1088/0264-9381/24/10/016
  * (20) H. Quevedo, in _Gravitation and Cosmology_ , _American Institute of Physics Conference Series_ , vol. 758, ed. by A. Macias, C. Lämmerzahl, D. Nunez (2005), _American Institute of Physics Conference Series_ , vol. 758, pp. 129–136.  DOI 10.1063/1.1900512.  ArXiv:gr-qc/0506139v1
  * (21) A. Ranada, Foundations of Physics 34(12), 1955 (2004).  DOI 10.1007/s10701-004-1629-y
  * (22) J. Brownstein, J. Moffat, Classical and Quantum Gravity 23(10), 3427 (2006).  DOI 10.1088/0264-9381/23/10/013
  * (23) H. Asada, Physics Letters B 661(2-3), 78 (2008).  DOI 10.1016/j.physletb.2008.02.006
  * (24) C. Massa, Astrophysics and Space Science 317(1-2), 139 (2008).  DOI 10.1007/s10509-008-9862-z
  * (25) R. Saffari, S. Rahvar, Physical Review D 77(10), 104028 (2008).  DOI 10.1103/PhysRevD.77.104028
  * (26) J.H. Gundlach, S. Schlamminger, C.D. Spitzer, K.Y. Choi, B.A. Woodahl, J.J. Coy, E. Fischbach, Physical Review Letters 98(15), 150801 (2007).  DOI 10.1103/PhysRevLett.98.150801
  * (27) G. Allemandi, M.L. Ruggiero, General Relativity and Gravitation 39(9), 1381 (2007).  DOI 10.1007/s10714-007-0441-3
  * (28) Q. Exirifard, Classical and Quantum Gravity 26(2), 025001 (2009).  DOI 10.1088/0264-9381/26/2/025001
  * (29) M.T. Jaekel, S. Reynaud, Classical and Quantum Gravity 22(11), 2135 (2005).  DOI 10.1088/0264-9381/22/11/015
  * (30) M.T. Jaekel, S. Reynaud, Classical and Quantum Gravity 23(3), 777 (2006).  DOI 10.1088/0264-9381/23/3/015
  * (31) M.T. Jaekel, S. Reynaud, Classical and Quantum Gravity 23(24), 7561 (2006).  DOI 10.1088/0264-9381/23/24/025
  * (32) K. Tangen, Physical Review D 76(4), 042005 (2007).  DOI 10.1103/PhysRevD.76.042005.
  * (33) C.M. Will, _Theory and experiment in gravitational physics_ , $2^{nd}$ edn. (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1993)
  * (34) C.M. Will, Living Reviews in Relativity 9(3), 1 (2006).  Mode of access: http://www.livingreviews.org/lrr-2006-3. — Date of access: 22.05.2006.
  * (35) R.C. Tolman, _Relativity, thermodynamics and cosmology_ (Oxford, Clarendon Press, 1934)
  * (36) H.J. Fahr, M. Siewert, ArXiv General Relativity and Quantum Cosmology e-prints arXiv:gr–qc/0610034v1 (2006)
  * (37) M. Mizony, M. Lachieze-Rey, Astronomy and Astrophysics 434, 45 (2005).  DOI 10.1051/0004-6361:20042195
  * (38) A.L. Licht, ArXiv General Relativity and Quantum Cosmology e-prints arXiv:gr-qc/0102103v1 (2001)
  * (39) C. Misner, K. Thorne, J. Wheeler, _Gravitation_ (W.H. Freeman, San Francisco, U.S.A., 1973)
  * (40) P.J.E. Peebles, B. Ratra, RMP 75(2), 559 (2003).  DOI 10.1103/RevModPhys.75.559.  arXiv:astro-ph/0207347v1
  * (41) K.P. Stanjukovich, V.N. Mel’nikov, _Hydrodynamics, fields and constants in the gravitational theory_ (E’nergoatomizdat, 1983).  (in Russian)
  * (42) E.V. Pitjeva, Astronomy Letters 31(3), 340 (2005)
  * (43) E.M. Standish, in _American Institute of Physics Conference Series_ , _American Institute of Physics Conference Series_ , vol. 977 (2008), pp. 254–263.  DOI 10.1063/1.2902789
  * (44) SAGAS Collaboration, ArXiv e-prints arXiv:0711.0304v3 [gr-qc] (2007)
  * (45) LATOR Collaboration, ArXiv General Relativity and Quantum Cosmology e-prints arXiv:gr-qc/0506104v3 (2005)
  * (46) The Pioneer Explorer Collaboration, ArXiv General Relativity and Quantum Cosmology e-prints arXiv:gr-qc/0506139v1 (2005)


# Tidal Deformability of Quark Stars with Repulsive Interactions

M. B. Albino† , R. Fariello† ‡ and F. S. Navarra  † Instituto de Física,
Universidade de São Paulo
Rua do Matão, 1371, 05508-090, Cidade Universitária, São Paulo, SP, Brazil  ‡
Departamento de Ciências da Computação, Universidade Estadual de Montes
Claros, Avenida Rui Braga, sn, Vila Mauricéia, Montes Claros, 39401-089, Minas
Gerais, Brazil

###### Abstract

In an early work, we applied a QCD-based EOS to the study of the stellar
structure of self-bound strange stars, obtaining sequences with maximum masses
larger than two solar masses and radii ranging from 8 to 12 Km. In this work,
we update the previous calculations and compare them with the most recent
data, including the very recent determination of the mass and radius of the
massive pulsar PSR J0740+6620 performed by the NICER and XMM-Newton
Collaborations. Our equation of state is similar to the MIT bag model one, but
it includes repulsive interactions, which turn out to be essential to
reproduce the accumulated experimental information. We find that our EOS is
still compatible with all astrophysical observations but the parameter window
is now narrower.

###### pacs:

PACS Numbers : 97.10.Cv, 12.38.-t, 12.38.Mh

##  I Introduction

The region of the QCD phase diagram with low temperature and high chemical
potential is still not well understood. From the theoretical point of view,
there are no accurate first principles predictions for the properties of QCD
matter at high baryon densities. The numerical lattice simulation techniques
that have been successfully applied to the study of the hot quark gluon plasma
(QGP) fail in the cold, baryon rich conditions due to the sign problem. In
spite of the difficulties, significant progress has been accomplished both in
the theoretical description of moderate density nuclear matter eosnuc ; tkhs
and ultrahigh-density matter gorda21 ; vuku20  but no reliable results exist
in the crucial regime between approximately one and ten nuclear saturation
densities. Several model calculations suggest that there is a low temperature
deconfined phase of quarks and gluons, the cold QGP, also called quark matter
(QM). This phase might exist in the the core of dense stars, an idea that has
been around already for some decades decs ; alf . It is even possible that a
whole star, not only its core, be made of quark matter witten84 . This
possibility was explored in several works qstars ; jorge20 ; fran12 ; jorge11
; pagliara2011  and will be further explored in this work, which is an update
of fran12 .

An early analysis of the existing observational data presented in ozel2006
concluded that most of the QM EOSs were too soft and therefore unable to
support the existence of neutron stars with a quark phase. Since then it was
shown in several works that a self-bound star, composed entirely of quark
matter, could explain a massive neutron star. In order to obtain a stiff
enough quark matter equation of state, several groups introduced repulsive
interactions among the quarks, mediated by the exchange of vector particles
fran12 ; vbag ; baym19 ; deb20 ; oert20 ; pisa21 ; sylhz , which can be
“effective massive gluons” or “effective vector mesons”. Interestingly, most
of these developments make use of a mean field approximation for the vector
field and arrive at a similar result, which is a quadratic term in the baryon
density present both in the pressure and energy density.

From the experimental side, during the last decade we have witnessed
remarkable advances in the observation of neutron stars: the discovery of
extremely massive neutron stars demo10 ; antoniadis ; qualitative improvements
in X-ray radius measurements gui13 ; oz16 ; freire16 ; stein17 ; stein16 ;
miller17 ; bogda16 ; and the famous LIGO/Virgo detection of gravitational
waves (GWs) originating from the NS-NS merger GW170817 ligo17 . Increasingly
stringent constraints have been placed on the EOS of NS matter. An accurate
measurement of a compact object using Shapiro delay croma  yielded
$2.14^{+0.1}_{-0.09}\,\,M_{\odot}$ for the J0740+6620 pulsar. It has been
argued that a handful of compact stars may achieve masses greater than the PSR
J0740+6620. The events denoted as GW190814 ligo20  and GW190425 ligo20a
suggest that the NS mass can be larger than $2.5\,\,M_{\odot}$. Recent data
(including a reliable determination of the radius) about the pulsar PSR
J0030+0451 were published in Refs. nicer1 ; nicer2 ; nicer3 ; nicer4 ; nicer5
. Finally, very recently nicer6  the NICER and XMM-Newton Collaborations
presented a determination of the mass and radius of the massive pulsar PSR
J0740+6620.

Differences between candidate EOSs can have a significant effect on the tidal
interactions of neutron stars. Recently new constraints appeared on the tidal
deformability ligo18 . It has been realized annala  that the two-solar-mass
constraint forces the EOS to be relatively stiff at low densities. At the same
time, the constraint on $\Lambda(1.4\,M_{\odot})$ sets an upper limit for the
stiffness, constraining the EOS band in a complementary direction.

In this paper we will update the study presented in fran12  and check whether
the EOS introduced in davi  remains a viable option, satisfying the most
recent experimental contraints.

This text is organized as follows. In Sec. II we briefly review the EOS for
the cold QGP. In Sec. III we introduce the stability conditions and discuss
its consequences. In Sec. IV we present the Tolman-Oppenheimer-Volkoff (TOV)
equations for stellar structure calculations and their numerical solutions. In
Sec. V we discuss the tidal deformability and in Sec. VI we present some
comments and conclusions.

##  II The equation of state

Following fran12 , we consider a quark star consisting of u, d and s quarks
with masses $m_{u}=$ $5$ MeV, $m_{d}=$ $7$ MeV, and $m_{s}=$ $100$ MeV. The
derivation of the EOS davi  used here starts with the assumption that the
gluon field can be decomposed into low (“soft”) and high (“hard”) momentum
components. The expectation values of the soft fields were identified with the
gluon condensates of dimension two and four, respectively. The former
generates a dynamical mass, $m_{G}$ for the hard gluons, and the latter yields
an analogue of the “bag constant” term in the energy density and pressure.
Given the large number of quark sources, even in the weak coupling regime, the
hard gluon fields are strong, the occupation numbers are large, and therefore
these fields can be approximated by classical color fields. The effect of the
condensates is to soften the EOS whereas the hard gluons significantly stiffen
it, by increasing both the energy density and pressure. With these
approximations it was possible to derive davi  an analytical expression for
the EOS, called MFTQCD (Mean Field Theory of QCD). When adapting this equation
of state to the stellar medium, we assume, as usual, that quarks and electrons
are in chemical equilibrium maintained by the weak processes farhi . Neutrinos
are assumed to escape and do not contribute to the pressure and energy
density. Moreover, we impose charge neutrality and baryon number conservation.
These requirements yield a set of four algebraic equations for Fermi momentum
calculation for each quark flavor ($u$, $d$ and $s$) and for the electrons
($e$)

${k_{u}}^{3}+{k_{d}}^{3}+{k_{s}}^{3}=3\pi^{2}\rho_{B},$
$2{k_{u}}^{3}={k_{d}}^{3}+{k_{s}}^{3}+{k_{e}}^{3},$ (1)
${k_{d}}^{2}+{m_{d}}^{2}={k_{s}}^{2}+{m_{s}}^{2},$
$\sqrt{{k_{u}}^{2}+{m_{u}}^{2}}+\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}=\sqrt{{k_{s}}^{2}+{m_{s}}^{2}},$

for a fixed baryon density $\rho_{B}$. The energy density is given by davi

$\epsilon=\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{2}}}\bigg{)}\
{\rho_{B}}^{2}+\mathcal{B}_{QCD}$
$+\sum_{i=u,d,s}3{\frac{\gamma_{Q}}{2{\pi}^{2}}}\Bigg{\\{}{\frac{{k_{i}}^{3}\sqrt{{k_{i}}^{2}+{m_{i}}^{2}}}{4}}+{\frac{{m_{i}}^{2}{k_{i}}\sqrt{{k_{i}}^{2}+{m_{i}}^{2}}}{8}}-{\frac{{m_{i}}^{4}}{8}}ln\Big{[}{k_{i}}+\sqrt{{k_{i}}^{2}+{m_{i}}^{2}}\
\Big{]}+{\frac{{m_{i}}^{4}}{16}}ln({m_{i}}^{2})\Bigg{\\}}$
$+{\frac{\gamma_{e}}{2{\pi}^{2}}}\Bigg{\\{}{\frac{{k_{e}}^{3}\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}}{4}}+{\frac{{m_{e}}^{2}{k_{e}}\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}}{8}}-{\frac{{m_{e}}^{4}}{8}}ln\Big{[}{k_{i}}+\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}\
\Big{]}+{\frac{{m_{e}}^{4}}{16}}ln({m_{e}}^{2})\Bigg{\\}},$ (2)

and the pressure is

$p=\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{2}}}\bigg{)}\
{\rho_{B}}^{2}-\mathcal{B}_{QCD}$
$+\sum_{i=u,d,s}{\frac{\gamma_{Q}}{2{\pi}^{2}}}\Bigg{\\{}{\frac{{k_{i}}^{3}\sqrt{{k_{i}}^{2}+{m_{i}}^{2}}}{4}}-{\frac{3{m_{i}}^{2}{k_{i}}\sqrt{{k_{i}}^{2}+{m_{i}}^{2}}}{8}}+{\frac{3{m_{i}}^{4}}{8}}ln\Big{[}{k_{i}}+\sqrt{{k_{i}}^{2}+{m_{i}}^{2}}\
\Big{]}-{\frac{3{m_{i}}^{4}}{16}}ln({m_{i}}^{2})\Bigg{\\}}$
$+{\frac{\gamma_{e}}{6{\pi}^{2}}}\Bigg{\\{}{\frac{{k_{e}}^{3}\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}}{4}}-{\frac{3{m_{e}}^{2}{k_{e}}\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}}{8}}+{\frac{3{m_{e}}^{4}}{8}}ln\Big{[}{k_{e}}+\sqrt{{k_{e}}^{2}+{m_{e}}^{2}}\
\Big{]}-{\frac{3{m_{e}}^{4}}{16}}ln({m_{e}}^{2})\Bigg{\\}},$ (3)

where $m_{e}=$ $0.5$ MeV is the electron mass, $m_{G}$ is the dynamical gluon
mass, and $g$ is the coupling constant $(\alpha_{s}=g^{2}/4\pi)$ in QCD. Our
analogue of the bag constant, called here $\mathcal{B}_{QCD}$, is given by

$\mathcal{B}_{QCD}=\frac{9}{128}\,\phi_{0}^{4}=\langle\frac{1}{4}F^{a\mu\nu}F^{a}_{\mu\nu}\rangle,$
(4)

where $\phi_{0}$ is an energy scale associated with the energy density of the
vacuum and with the gluon condensate davi . In (2) and (3) the summation over
quark colors has already been performed. Throughout this work we employ the
natural units $G=1$, $\hbar=1$, $c=1$. Comparing Eqs. (2) and (3) with the
equivalent definitions of energy and pressure in the modified bag model with
postulated repulsive vector interactions (see Eqs. (14) and (16) of Ref. bla20
), we observe a similarity. Both EOSs have a term proportional to
$\rho_{B}^{2}$. In davi  it was derived from QCD whereas in bla20  it was
postulated.

##  III Stability conditions

In this section we discuss the two stability conditions, which have to be
satisfied by stable strange quark matter. The first one is that the energy per
baryon of the deconfined phase (for $P=0$ and $T=0$) is lower than the
nonstrange infinite baryonic matter defined in farhi ; pagliara2011 .
Following these works we impose that:

$E_{A}\equiv\frac{\epsilon}{\rho_{B}}\leq 934\,\,\,\mbox{MeV}.$ (5)

This condition must hold at the zero pressure point and hence we can, from (2)
and (3), numerically derive a relation between the bag constant $B_{QCD}$ and
the ratio $\xi=g/m_{G}$. We solve (3) obtaining
$\rho_{B}=\rho_{B}(B_{QCD},\xi)$, which is then inserted into (2). The
resulting expression is used to write the condition
$\epsilon(B_{QCD},\xi)/\rho_{B}(B_{QCD},\xi)=934$ MeV, which defines one
“stability frontier”. This last equation is rewritten as $\xi=\xi(B_{QCD})$,
is plotted in Fig. 1 (solid line) and denoted by the 3-flavor line. Points in
the $(\mathcal{B}_{QCD},\xi)$ plane located on the right of the solid line are
discarded since they do not satisfy (5). The solid line, corresponding to the
maximal value of $E_{A}=934$ MeV, determines the maximum value of
$\mathcal{B}_{QCD}$. The minimum value of $\mathcal{B}_{QCD}$ is determined by
the second stability condition, which requires nonstrange quark matter in the
bulk to have an energy per baryon higher than the one of nonstrange infinite
baryonic matter. By imposing that

$E_{A}\equiv\frac{\epsilon}{\rho_{B}}\geq 934\,\,\,\mbox{MeV},$ (6)

for a two flavor quark matter at ground state, we ensure that atomic nuclei do
not dissolve into their constituent quarks. The constraint (6) defines the
dashed line in the $(\mathcal{B}_{QCD},\xi)$ plane, denoted by the 2-flavor
line in Fig. 1. Points located on the left of this line are excluded because
they do not satisfy (6). The region between the two lines in Fig. 1 defines
our stability window.

Figure 1: Values of $\xi=g/m_{G}$ as a function of $\mathcal{B}_{QCD}$ for
different values of the energy per baryon. The two lines define the stability
region.

Having fixed the $\mathcal{B_{QCD}}$ and $\xi$ parameters, we go back to (2)
and (3) and, obtaining $\epsilon$ and $p$ for successive values of $\rho_{B}$,
we construct the EOS in the form $p=p(\epsilon)$, plotted in Fig. 2a. In the
figure, the different lines correspond to the three parameter sets listed in
Table I. In this type of plot the slope is the speed of sound, which, due to
causality, can not exceed the unity. This limit is shown by the full lines in
the figure. In Fig. 2b we show the corresponding values of the speed of sound.
As it can be seen, our model yields a much stiffer EOS, with a speed of sound
much larger than the conformal value, for which $c^{2}_{s}=1/3$. The dot-
dashed line shows the EOS obtained from a recently updated version of the MIT
bag model zzl ; german , which reads

$p(\epsilon)=\frac{(\epsilon-
B_{eff})}{3}-\frac{a_{2}^{2}}{12\pi^{2}a_{4}}\left[1+\sqrt{1+\frac{16\pi^{2}a_{4}}{a_{2}^{2}}(\epsilon-
B_{eff})}\right],$ (7)

where $B_{eff}^{1/4}=142.52$ MeV, $a_{2}^{1/2}=100$ MeV and $a_{4}=0.535$. As
it can be seen, the MFTQCD EOS generates stronger pressure for larger values
of the parameter $\xi=g/m_{G}$. This combination of parameters appears in the
first term of (3), which comes from the repulsive interactions davi .

|  |
---|---|---
(a) |  | (b)
Figure 2: a) Equation of state obtained with MFTQCD. Set I, II and III
correspond to the parameter combinations shown in Table I. For comparison, the
dot-dashed line show the MIT Bag Model EOS used by Parisi et al. in Ref.
german . b) Speed of sound for the same parameter choices.

##  IV TOV equation, mass and radius

In order to describe the structure of a static, non-rotating compact star, we
use the Tolman-Oppenheimer-Volkoff (TOV) equation for the pressure $p(r)$
glend :

$\frac{dp}{dr}=-\frac{\epsilon(r)M(r)}{r^{2}}\left[1+\frac{p(r)}{\epsilon(r)}\right]\left[1+\frac{4\pi
r^{3}p(r)}{M(r)}\right]\times\left[1-\frac{2M(r)}{r}\right]^{-1}.$ (8)

The enclosed mass $M(r)$ of the compact star is given by the mass continuity
equation:

$\frac{dM(r)}{dr}=4\pi r^{2}\epsilon(r).$ (9)

Equations (8) and (9) express the balance between the gravitational force and
the internal pressure acting on a shell of mass $dM(r)$ and thickness $dr$.

We solve numerically (8) and (9) for $p(r)$ and $M(r)$, to obtain the mass-
radius diagram. The pressure and the energy density in (8) and (9) are given
by the MFTQCD expressions (3) and (2), respectively. We take the central
energy density to be $\epsilon(r=0)=\epsilon_{c}$ and then we integrate out
(8) and (9) from $r=0$ up to $r=R$, where the pressure at the surface is zero:
$p(r=R)=0$. In Fig. 3 we show the mass-radius diagram for several values of
$\mathcal{B_{QCD}}$ and $\xi$ respecting the stability condition.

Figure 3: Mass-radius diagram for combinations of $\mathcal{B_{QCD}}$ and
$\xi$ allowed by the stability conditions. Set I, II and III correspond to the
parameter combinations shown in Table I. The points represent the region
favored by the measurements reported by the NICER and XMM-Newton
Collaborations nicer1 ; nicer2 ; nicer3 ; nicer4 ; nicer5 ; nicer6 . The
horizontal line shows the mass of the compact object observed in the event
GW190814.

In the diagram, the points represent the region favored by the measurements
reported in Refs. nicer1 ; nicer2 ; nicer3 ; nicer4 ; nicer5 ; nicer6 . We can
see that, with the parameters chosen in the indicated range, our EOS is able
to satisfy all the constraints shown in the mass-radius diagram.

Table 1: Parameter sets used in the figures. Set | $\mathcal{B_{QCD}}(MeV/fm^{3})$ | $\xi(MeV^{-1})$
---|---|---
I | 70 | 0.0011
II | 60 | 0.0016
III | 50 | 0.0022

##  V Tidal deformability

An object that experiences the tidal force of another object will deform. The
susceptibility to deform is often measured using dimensionless quantities that
are called Love numbers. The Love number is an interesting quantity because it
can be used to probe the dense-matter EOS using data from double-neutron-star-
merger events. There are various binary systems where two objects orbit around
each other. Because these objects loose energy to gravitational waves, their
orbits are not stable. Therefore, they will inevitably approach each other
until they finally merge. Around the collision point, the generated
gravitational-wave signal is strong enough to be detected by terrestrial
instruments.

The tidal deformability parameter is given by tanja ; tanja2 ; sabatucci

$\Lambda=\frac{2}{3}k_{2}C^{-5},$ (10)

where $C\equiv M/R$ is the compactness of the star and $k_{2}$ is the tidal
Love number, which is given by tanja ; tanja2 ; sabatucci

$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{8C^{5}}{5}\left(1-2C\right)^{2}\left[2+2C\left(y-1\right)-y\right]\left\\{2C\left[6-3y+3C\left(5y-8\right)\right]\right.$
(11)
$\displaystyle\quad+4C^{3}\left[13-11y+C\left(3y-2\right)+2C^{2}\left(1+y\right)\right]$
$\displaystyle\quad\left.+3\left(1-2C\right)^{2}\left[2-y+2C\left(y-1\right)\right]\ln\left(1-2C\right)\right\\}^{-1},$

where

$y=\frac{R\,\beta(R)}{H(R)}-\frac{4\pi
R^{3}\epsilon_{sup}}{M},\quad\beta(r)=\frac{dH(r)}{dr}.$ (12)

In the above equation, the second term is a correction due to the fact that in
our model the energy density at the surface of the star,
$\epsilon_{sup}\equiv\epsilon(P=0)$ is not zero tanja2 . The functions $H$ and
$\beta$ can be obtained by solving the following system of differential
equations:

$\displaystyle H^{\prime}(r)$ $\displaystyle=$ $\displaystyle\beta(r),$ (13)
$\displaystyle\frac{d\beta}{dr}$ $\displaystyle=$ $\displaystyle
2\left(1-\frac{2M(r)}{r}\right)^{-1}H(r)\left\\{-2\pi\left[5\epsilon(r)+9P(r)+\frac{\epsilon(r)+P(r)}{dP/d\epsilon}\right]\right.$
(14)
$\displaystyle\left.\quad+\frac{3}{r^{2}}+2\left(1-\frac{2M(r)}{r}\right)^{-1}\left(\frac{M(r)}{r^{2}}+4\pi
rP(r)\right)^{2}\right\\}$
$\displaystyle\quad+\frac{2\beta(r)}{r}\left(1-\frac{2M(r)}{r}\right)^{-1}\left[-1+\frac{M(r)}{r}+2\pi
r^{2}\left(\epsilon(r)-P(r)\right)\right].$

The Love number $k_{2}$ measures how easily the bulk of the matter in a star
is deformed. The Love number also encodes information about the star’s degree
of central condensation. Stars that are more centrally condensed will have a
smaller response to a tidal field, resulting in a smaller Love number. The
Love number decreases with increasing compactness, and from Eq. (11) it can be
seen that $k_{2}$ vanishes at the compactness of a black hole ($M/R=0.5$)
regardless of the EOS dependent quantity $y$. The tidal Love numbers of
strange quark matter stars are qualitatively different from those of hadronic
matter stars tanja2 ; latt10 ; under20 . The latter decrease strongly for
small values of the compactness.

In Fig. 4a we show the Love number $k_{2}$ as a function of the compactness
$C$. We expect that a very compact star for any EoS is harder to deform then a
less compact one. This is what we see in the figures. It is interesting to
observe that the same variation of $\mathcal{B_{QCD}}$ and $\xi$ which
produces visible effects in the equation of state and in the mass-radius
diagram does not lead to appreciable differences in the $k_{2}$-$C$ plot. The
curves shown in Fig. 4a are practically identical to the curves in the
analogous plots shown in Refs. tanja2 , latt10  and under20 , which were
obtained with strange quark matter equations of state. This suggests that a
wide variety of quark matter EOSs lead to the same values of $k_{2}$. We also
note that our curves are close to the one obtained with the ultrarelativistic
EOS with the speed of sound $c^{2}_{s}=1/3$ under20 . For completeness, in
Fig. 4b we show the Love number $k_{2}$ as a function of the variable $y$.

|  |
---|---|---
(a) |  | (b)
Figure 4: a) Tidal Love number $k_{2}$ as a function of the compactness. b)
$k_{2}$ as a function of $y$. The different lines correspond to the three
parameter sets listed in Table I.

As pointed out in tanja2 , in contrast to the Love number, the tidal
deformability has a wide range of values, spanning roughly an order of
magnitude over the observed mass range of neutron stars in binary systems. The
updated version of the tidal deformability estimate for a $1.4\,M_{\odot}$
neutron star based on the gravitational-wave event GW170817 ligo18  implies
that

$70<\Lambda_{1.4}<580.$ (15)

In Fig. 5 we show our results for $\Lambda$ as a function of the star mass
$M$. As it can be seen, the constraint (15) can be satisfied. We note,
however, the visible tension between this constraint and those shown in the
mass-radius plot. The larger values of the radius required to fit the NICER
points seem to be somewhat difficult to reconcile with the $\Lambda$ values
required by the GW170817 estimates. Other calculations performed with quark
matter stars laura ; zzl ; wsz ; zm ; lz  or hybrid stars sylhz ; consta ;
juwu ; theo ; kmt ; han18  arrive at similar results. On the other hand,
calculations of the tidal deformability with purely hadronic equations of
states latt18 ; tan ; latt21  seem to reproduce the experimental data more
easily.

Figure 5: The tidal deformability parameter $\Lambda$ as a function of the
star mass. The different lines correspond to the three parameter sets listed
in Table I. The vertical bar is the empirical tidal deformability at
$M=1.4M_{\odot}$ inferred from the Bayesian analysis of the GW170817 data at
the 90% confidence level ligo18 .

##  VI Conclusion

In davi  a new equation of state for cold quark matter was presented. It was
soon applied to the the study of neutron stars, treated as self-bound strange
quark stars. In this paper, almost ten years later, we have updated the
calculations published in fran12  and checked whether that EOS can still
account for the most recent astrophysical data. We find that MFTQCD is still a
viable option. However, we observe that the parameter window is closing. A
confirmation of the existing data and the reduction of the error bars in the
tidal deformability and in the NICER neutron star radii data will be crucial
to rule out strange quark star models and reduce the freedom in the choice of
the equation of state.

###### Acknowledgements.

We are deeply grateful to J. Horvath and to G. Lugones for fruitful
discussions. This work was partially financed by the Brazilian funding
agencies CAPES and CNPq.

## References

  * (1) M. Oertel, M. Hempel, T. Klähn and S. Typel, Rev. Mod. Phys. 89, 015007 (2017).
  * (2) I. Tews, T. Krüger, K. Hebeler and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013).
  * (3) T. Gorda, A. Kurkela, R. Paatelainen, S. Säppi and A. Vuorinen, arXiv:2103.07427 [hep-ph].
  * (4) E. Annala, T. Gorda, A. Kurkela, J. Nättilä and A. Vuorinen, Nature Phys. 16, 907 (2020).
  * (5) D. Ivanenko and D. F. Kurdgelaidze, Astrophysics J. 1, 251 (1965); D. Ivanenko and D. F. Kurdgelaidze, Lett. Nuovo Cimento 2, 13 (1969); N. Itoh, Prog. Theor. Phys. 44, 291 (1970).
  * (6) M. Alford, D. Blaschke, A. Drago, T. Klahn, G. Pagliara, and J. Schaffner-Bielich, Nature 445, E7 (2007).
  * (7) E. Witten, Phys. Rev. D 30, 272 (1984).
  * (8) F. Weber, Prog. Part. and Nucl. Phys. 54, 193 (2005); R. X. Xu, S. I. Bastrukov, F. Weber, J. W. Yu, and I. V. Molodtsova, Phys. Rev. D 85, 023008 (2012); A. Drago, A. Lavagno, and I. Parenti, Astrophys. J. 659, 1519 (2007); J. W. Yu and R. X. Xu, Mon. Not. R. Astron. Soc. 414, 489 (2011); A. Drago, A. Lavagno, and G. Pagliara, Phys. Rev. D 89, 043014 (2014); A. Drago and G. Pagliara, Phys. Rev. C 92, 045801 (2015); A. Drago, A. Lavagno, G. Pagliara, and D. Pigato, Eur. Phys. J. A 52, 40 (2016); A. Drago and G. Pagliara, Eur. Phys. J. A 52, 41 (2016); G. Panotopoulos and I. Lopes, Phys. Rev. D 96, 083013 (2017); G. Wiktorowicz, A. Drago, G. Pagliara, and S. B. Popov, Astrophys. J. 846, 163 (2017); J. Bora and U. D. Goswami, Mon. Not. Roy. Astron. Soc. 502, 1557 (2021).
  * (9) J. E. Horvath and P. H. R. S. Moraes, Int. J. Mod. Phys. D 30, 2150016 (2021).
  * (10) B. Franzon, D. A. Fogaca, F. S. Navarra and J. E. Horvath, Phys. Rev. D 86, 065031 (2012).
  * (11) M.G.B. de Avellar, J. E. Horvath, and L. Paulucci, Phys. Rev. D 84, 043004 (2011).
  * (12) S. Weissenborn, I. Sagert, G. Pagliara, M. Hempel, and J. Schaffner - Bielich, Astrophysics J. 740, L14 (2011).
  * (13) F. Özel, Nature 441, 1115 (2006).
  * (14) M. Cierniak, T. Klähn, T. Fischer and N. U. Bastian, Universe 4, 30 (2018); T. Klähn and T. Fischer, Astrophys. J. 810, 134 (2015).
  * (15) Y. Song, G. Baym, T. Hatsuda and T. Kojo, Phys. Rev. D 100, 034018 (2019).
  * (16) L. L. Lopes, C. Biesdorf, K. D. Marquez and D. P. Menezes, Phys. Scripta 96, 065302 (2021). L. L. Lopes, C. Biesdorf and D. P. Menezes, arXiv:2005.13136 [hep-ph].
  * (17) K. Otto, M. Oertel and B. J. Schaefer, Eur. Phys. J. ST 229, 3629 (2020).
  * (18) R. D. Pisarski, Phys. Rev. D 103, L071504 (2021).
  * (19) L. Q. Su, Y. Yan, C. M. Li, Y. F. Huang and H. Zong, Phys. Rev. D 103, 094037 (2021).
  * (20) J. Antoniadis et al., Science 340, 6131 (2013).
  * (21) P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels, Nature 467, 1081 (2010).
  * (22) S. Guillot, M. Servillat, N. A. Webb and R. E. Rutledge, Astrophys. J. 772, 7 (2013).
  * (23) F. Ozel, D. Psaltis, T. Guver, G. Baym, C. Heinke and S. Guillot, Astrophys. J. 820, 28 (2016).
  * (24) F. Ozel and P. Freire, Ann. Rev. Astron. Astrophys. 54, 401 (2016).
  * (25) A. W. Steiner, C. O. Heinke, S. Bogdanov, C. Li, W. C. G. Ho, A. Bahramian and S. Han, Mon. Not. Roy. Astron. Soc. 476, 421 (2018).
  * (26) J. Nättilä, A. W. Steiner, J. J. E. Kajava, V. F. Suleimanov and J. Poutanen, Astron. Astrophys. 591, A25 (2016).
  * (27) J. Nättilä, M. C. Miller, A. W. Steiner, J. J. E. Kajava, V. F. Suleimanov and J. Poutanen, Astron. Astrophys. 608, A31 (2017).
  * (28) S. Bogdanov, C. O. Heinke, F. Özel and T. Güver, Astrophys. J. 831, 184 (2016).
  * (29) B. P. Abbott et al. [LIGO Scientic and Virgo Collaborations], Phys. Rev. Lett. 119, 161101 (2017).
  * (30) H. Cromartie et. al. Nature Astronomy 4, 72 (2020).
  * (31) R. Abbott et al. LIGO and Virgo Scientific Collaborations, Astrophys. J. Lett. 896, L44 (2020).
  * (32) B. P. Abbott et al. LIGO Scientific and Virgo Collaborations, Astrophys. J. Lett. 892, L3 (2020).
  * (33) M.C. Miller et al., Astrophys. J. Lett. 887, L24 (2019).
  * (34) T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot et al., Astrophys. J. Lett. 887, L21 (2019).
  * (35) S. Bogdanov, S. Guillot, P. S. Ray, M. T. Wolff, D. Chakrabarty, W. C. G. Ho et al., Astrophy. J. Lett. 887, L25 (2019).
  * (36) Raaijmakers, G. et al., Astrophys. J. Lett. 887, L22 (2019).
  * (37) C.D. Capano et al., Nature Astron. 4, 625 (2020).
  * (38) T. E. Riley et al., arXiv:2105.06980 [astro-ph.HE].
  * (39) R. Abbott et al. LIGO and Virgo Scientific Collaborations, Phys. Rev. Lett. 121, 161101 (2018).
  * (40) Annala, E., Gorda, T., Kurkela, A. and Vuorinen, A., Phys. Rev. Lett. 120, 172703 (2018).
  * (41) D. A. Fogaça, F.S. Navarra, Phys. Lett. B 700, 236 (2011).
  * (42) E. Farhi, R.L. Jaffe, Phys. Rev. D 30, 2379 (1984).
  * (43) M. Cierniak and D. Blaschke, Eur. Phys. J. ST 229, 3663 (2020).
  * (44) E. P. Zhou, X. Zhou and A. Li, Phys. Rev. D 97, 083015 (2018).
  * (45) A. Parisi, C. V. Flores, C. H. Lenzi, C. S. Chen and G. Lugones, arXiv:2009.14274.
  * (46) N. Glendenning, Compact stars, (Springer, New York, 2000).
  * (47) T. Hinderer, arXiv: 0711.2420; E. E. Flanagan and T. Hinderer, Phys. Rev. D 77, 021502 (2008); T. Hinderer, Astrophys. J. 677, 1216 (2008).
  * (48) T. Hinderer, B. D. Lackey, R. N. Lang, J. S. Read, Phys. Rev. D 81, 123016 (2010).
  * (49) A. Sabatucci Tidal deformation of neutron stars, Thesis, La Sapienza – University of Rome (2018).
  * (50) S. Postnikov, M. Prakash and J. M. Lattimer, Phys. Rev. D 82, 024016 (2010).
  * (51) A. Zacchi, arXiv:2007.00423.
  * (52) O. Lourenço, C. H. Lenzi, M. Dutra, E. J. Ferrer, V. de la Incera, L. Paulucci and J. E. Horvath, arXiv:2104.07825.
  * (53) Q. Wang, C. Shi and H. S. Zong, Phys. Rev. D 100, 123003 (2019); Erratum: [Phys. Rev. D 100, 129903 (2019)].
  * (54) C. Zhang and R. B. Mann, Phys. Rev. D 103, 063018 (2021).
  * (55) C. M. Li, S. Y. Zuo, Y. Yan, Y. P. Zhao, F. Wang, Y. F. Huang and H. S. Zong, Phys. Rev. D 101, 063023 (2020).
  * (56) M. Ferreira, R. Câmara Pereira and C. Providência, Phys. Rev. D 102, 083030 (2020).
  * (57) M. Ju, X. Wu, F. Ji, J. Hu and H. Shen, Phys. Rev. C 103, 025809 (2021).
  * (58) T. F. Motta, A. M. Kalaitzis, S. Antic, P. A. M. Guichon, J. R. Stone and A. W. Thomas, Astrophys. J. 878, 159 (2019).
  * (59) S. Khanmohamadi, H. R. Moshfegh and S. Atashbar Tehrani, Phys. Rev. D 101, 123001 (2020).
  * (60) S. Han and A. W. Steiner, Phys. Rev. D 99, 083014 (2019).
  * (61) T. Zhao and J. M. Lattimer, Phys. Rev. D 98, 063020 (2018).
  * (62) N. H. Tan, D. T. Khoa and D. T. Loan, Eur. Phys. J. A 57, 153 (2021).
  * (63) C. Drischler, S. Han, J. M. Lattimer, M. Prakash, S. Reddy and T. Zhao, Phys. Rev. C 103, 045808 (2021).