# Generalized uncertainty principle and neutrino phenomenology

Ioannis D. Gialamas  0000-0002-2957-5276<EMAIL_ADDRESS>Laboratory
of High Energy and Computational Physics, National Institute of Chemical
Physics and Biophysics, Rävala pst. 10, Tallinn, 10143, Estonia  Timo J.
Kärkkäinen  0000-0002-2885-2235<EMAIL_ADDRESS>Laboratory of High Energy
and Computational Physics, National Institute of Chemical Physics and
Biophysics, Rävala pst. 10, Tallinn, 10143, Estonia  Luca Marzola
0000-0003-2045-1100<EMAIL_ADDRESS>Laboratory of High Energy and
Computational Physics, National Institute of Chemical Physics and Biophysics,
Rävala pst. 10, Tallinn, 10143, Estonia
Institute of Computer Science, University of Tartu, Narva mnt 18, 51009 Tartu,
Estonia

###### Abstract

### Abstract

Generalized uncertainty principles are effective changes to the Heisenberg
uncertainty principle that emerge in several quantum gravity models. In the
present letter, we study the consequences that two classes of these
modifications yield on the physics of neutrinos. Besides analyzing the change
in the oscillation probabilities that the generalized uncertainty principles
entail, we assess their impact on the neutrino coherence length and their
possible interpretation as nonstandard neutrino interactions. Constraints cast
by present and planned neutrino experiments on the generalized uncertainty
principles parameters are also derived.

##  I Introduction

Research into the phenomenology of Generalized Uncertainty Principles (GUPs)
originated with the attempt to construct a coherent theory of quantum gravity
(QG)—still one of the most important goals of contemporary theoretical
physics. Despite extensive research in the last decades, any of the proposed
QG theories has yet to receive a first experimental validation as, generally,
current data does not yet probe the regime where the proposed flavour of
quantumness appears. The usual treatment of gravitational effects then
commonly relies on effective classical theories, believed to hold at energy
scales below the Planck mass, $M_{\rm Pl}\simeq 10^{19}\;\mathrm{GeV}$, where
the quantum regime should finally take over. Various QG theories, including
promising candidates such as string theory Amati  _et al._ (1987); Konishi
_et al._ (1990) and loop quantum gravity Rovelli and Smolin (1995), propose
that the onset of a quantum regime is accompanied by the emergence of a
minimum length comparable to the Planck scale $\sim 10^{-35}\,{\rm m}$
111Importantly, theories incorporating extra dimensions may support larger
gravitational constant values, thereby leading to a larger Planck length and,
consequently, a greater expected minimum length scale Arkani-Hamed  _et al._
(1998); Antoniadis  _et al._ (1998); Arkani-Hamed  _et al._ (1999). Regardless
of the specifics of the framework, the introduction of a minimum length
requires adjustments to the Heisenberg Uncertainty Principle which, in turn,
yield a GUP Maggiore (1993a); Garay (1995); Kempf  _et al._ (1995); Adler and
Santiago (1999); Chen and Adler (2003); Das and Vagenas (2008); Adler (2010).

Since their first appearance in QG theories, GUPs have been the subject of
dedicated studies aimed at exploring the related phenomenological consequences
within gravity theories Maggiore (1994, 1993b); Scardigli (1999), as well as
in many other fields of physics including, for instance, black holes Adler
_et al._ (2001); Myung  _et al._ (2007); Nozari and Hamid Mehdipour (2008);
Bargueño and Vagenas (2015); Carr  _et al._ (2015); Casadio  _et al._ (2018);
Alonso-Serrano  _et al._ (2018); Ong (2018); Gangopadhyay and Dutta (2018);
Mureika (2019); Knipfer  _et al._ (2019); Carr  _et al._ (2020); Zhou  _et
al._ (2022); Carr (2022) and gravitational waves physics Feng  _et al._
(2017); Bhattacharyya  _et al._ (2020); Diab and Tawfik (2022); Das  _et al._
(2021, 2022), with still continuing activity Benczik  _et al._ (2005); Bambi
and Urban (2008); Marin  _et al._ (2013); Scardigli and Casadio (2015);
Bernaldez  _et al._ (2023); Artigas  _et al._ (2024); Pachol and Wojnar
(2023); Kozak  _et al._ (2023); Bosso  _et al._ (2023a); Wojnar (2024);
Segreto and Montani (2024); Hong  _et al._ (2024); Ali and Wojnar (2024);
Scardigli  _et al._ (2017). We refer the reader to Ref. Bosso  _et al._
(2023b) for a recent review on the topic.

In this letter we continue the exploration of GUPs by investigating a possible
low energy phenomenology that yields testable effects on the properties of
neutrinos. These particles provide an ideal laboratory to test GUP scenarios
because the flavor oscillations observed at dedicated experiments are
inherently a quantum phenomenon and, as such, are directly sensitive to the
workings of quantum mechanics that GUPs modify. The work improves on previous
studies of GUPs in neutrino phenomenology Sprenger  _et al._ (2011); Torri and
Miramonti (2024) by considering different GUP formulations and updated results
of neutrino oscillation experiments. The assessment of the impact of GUPs on
the neutrino decoherence length and their interpretation as nonstandard
neutrino interactions are also novel results of the analysis. In the
following, prior to showing how GUPs modify the phenomenology of neutrinos, we
briefly review the GUP scenarios that will be considered in our study.

##  II The generalized uncertainty principle

Depending on the specific parametrization, modifications of the Heisenberg
uncertainty principle introduce in the theory a minimum measurable length
which can be accompanied by a maximum measurable momentum. In the following,
we briefly review each of these prototypical cases which we later use in our
analysis.

### GUP-I: minimum measurable length

The first GUP parametrization we consider is of the form Kempf  _et al._
(1995):

$\displaystyle\Delta x_{i}\Delta p_{i}\geq\frac{\hbar}{2}\big{[}$
$\displaystyle 1+\beta\left((\Delta p)^{2}+\langle p\rangle^{2}\right)$
$\displaystyle+2\beta\left((\Delta p_{i})^{2}+\langle
p_{i}\rangle^{2}\right)\big{]}\,,$ (1)

with $i=1,2,3$ and $p=\sqrt{\sum_{i}p_{i}p_{i}}$. The parameter $\beta$ is
dubbed GUP parameter, and it is defined as $\beta=\beta_{0}l_{\rm
Pl}^{2}/\hbar^{2}=\beta_{0}/(cM_{\rm Pl})^{2}$. In this context, $\beta_{0}$
is a dimensionless parameter that regulates the value of the minimum length
while $l_{\rm Pl}$ and $M_{\rm Pl}$ are the Planck length and mass,
respectively. In natural units, the GUP parameter is $\beta\simeq
7\beta_{0}\times 10^{-39}\;\mathrm{GeV}^{-2}$.

Upon minimizing (II) with respect to $\Delta p_{i}$, it becomes evident that
the GUP implies a minimum length $\Delta x_{i}^{\rm
min}\sim\sqrt{\beta_{0}}l_{\rm Pl}$ for the measurement of any spatial
Cartesian coordinate.

The GUP form under consideration arises from modifications of the Heisenberg
algebra, as elaborated in Kempf  _et al._ (1995). This altered algebra, which
serves as the foundation for the GUP, is expressed by the equation:

$[x_{i},p_{j}]=i\hbar\left(\delta_{ij}+\beta\delta_{ij}p^{2}+2\beta
p_{i}p_{j}\right)\,.$ (2)

It is worth noting that this particular form of the commutator ensures that
$[x_{i},x_{j}]=[p_{i},p_{j}]=0$. We remark that this property also applies to
the GUP formulation which we will discuss subsequently.

We use the transformations

$x_{i}=\tilde{x}_{i}\,,\qquad
p_{i}=\tilde{p}_{i}(1+\beta\tilde{p}^{2})\,,\qquad E/c=p_{0}=\tilde{p}_{0}\,,$
(3)

to define tilted variables that satisfy the canonical commutation relations
$[\tilde{x}_{i},\tilde{p}_{j}]=i\hbar\delta_{ij}$. The squared $4-$momentum of
the physical variables $p$ and $p_{0}$ is expanded as222Note that we have used
the mostly minus sign convention and from now on we set $c=1$.

$p_{0}^{2}-p^{2}=\tilde{p}_{0}^{2}-\tilde{p}^{2}\left(1+2\beta\tilde{p}^{2}+\mathcal{O}(\beta^{2})\right)\,,$
(4)

where the higher order terms in the $\beta-$expansion are Planck suppressed.
The tilded variables satisfy the canonical dispersion relation, namely
$\tilde{p}_{0}^{2}-\tilde{p}^{2}=m^{2}$, so using this along with the inverse
transformation $\tilde{p}^{2}=p^{2}\left(1-2\beta
p^{2}+\mathcal{O}(\beta^{2})\right)$, Eq. (4) can be rewritten as

$E^{2}=m^{2}+p^{2}(1-2\beta p^{2})\,,$ (5)

where for $\beta\rightarrow 0$ the standard energy-momentum dispersion
relation is restored. At the leading order in the GUP parameter we have

$\Delta p=\Delta E+3\beta E^{2}\Delta E+\mathcal{O}(\beta^{2})\,.$ (6)

### GUP-II: minimum measurable length and maximum measurable momentum

Next, we examine the generalization suggested in Ali  _et al._ (2009); Das
_et al._ (2010), wherein the commutation relation satisfied by the modified
position and momentum is given by

$[x_{i},p_{j}]=i\hbar\left(\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}+\alpha^{2}\left(p^{2}\delta_{ij}+3p_{i}p_{j}\right)\right)\right)\,,$
(7)

where $\alpha=\alpha_{0}l_{\rm Pl}/\hbar$ is the “GUP-II parameter”. This
formulation of GUP implies a minimum measurable length $\Delta x_{i}^{\rm
min}\sim\alpha_{0}l_{\rm Pl}$ and a maximum measurable momentum $\Delta
p_{i}^{\rm max}\sim M_{\rm Pl}/\alpha_{0}$. The corresponding dispersion
relation is333The momentum transformation (3) in this case takes the form
$p_{i}=\tilde{p}_{i}(1-\alpha\tilde{p}_{i}+2\alpha^{2}\tilde{p}_{i}^{2})$.
This yields a modified dispersion relation of the form
$E^{2}=m^{2}+p^{2}(1\pm\alpha p)^{2}$. In our analysis, we retain only the
minus sign, as the alternative leads to superluminal speed of gravitational
waves Das  _et al._ (2021).

$E^{2}=m^{2}+p^{2}(1-\alpha p)^{2}\,.$ (8)

Expanding again at the leading order in the relevant GUP parameter we find

$\Delta p=\Delta E+2\alpha E\Delta E+\mathcal{O}(\alpha^{2})\,.$ (9)

Although QC models predict specific value of $\alpha$ or $\beta$, in the
following we consider arbitrary values for the parameters in order to assess
the power of neutrino experiments to constrain GUP scenarios.

##  III Experimental constraints and prospects from neutrino oscillations

Prior to quantifying the impact of the modified dispersion relations in Eqs.
(5) and (8) on neutrino oscillations, we briefly summarize the workings of
this effect.

###  III.1 Standard neutrino oscillations

The misalignment between neutrino mass and flavour (interaction) eigenstates
is described by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing
matrix $U$:

$\begin{pmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{pmatrix}=\begin{pmatrix}U_{e1}&U_{e2}&U_{e3}\\\ U_{\mu
1}&U_{\mu 2}&U_{\mu 3}\\\ U_{\tau 1}&U_{\tau 2}&U_{\tau
3}\end{pmatrix}\begin{pmatrix}\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\end{pmatrix}\,.$
(10)

The matrix elements depend on three mixing angles $\theta_{12}$,
$\theta_{13}$, $\theta_{23}$ and a CP violating phase $\delta$, which later we
set to the experimental values provided by the NuFit collaboration Esteban
_et al._ (2020) in our numerical analysis.

Let a neutrino be produced in a pure state at $t=0$ by a Standard Model (SM)
gauge interaction. The time evolution of its flavour state is given by

$\nu_{\ell}(t)=\sum_{\begin{subarray}{c}\ell^{\prime}=e,\mu,\tau\\\
i=1,2,3\end{subarray}}U_{\ell
i}U_{\ell^{\prime}i}^{*}e^{-iE_{i}t/\hbar}\nu_{\ell^{\prime}}(0)\equiv\sum_{\ell^{\prime}=e,\mu,\tau}A_{\ell\ell^{\prime}}\nu_{\ell^{\prime}}(0)\,,$
(11)

where $A_{\ell\ell^{\prime}}$ is the $\ell\rightarrow\ell^{\prime}$ flavour
transition probability amplitude at time $t>0$. The corresponding transition
probability is

$P_{\ell\ell^{\prime}}=|A_{\ell\ell^{\prime}}|^{2}=\sum\limits_{i,j=1}^{3}U_{\ell
i}^{*}U_{\ell^{\prime}i}U_{\ell
j}U_{\ell^{\prime}j}^{*}e^{-i(E_{i}-E_{j})t/\hbar}\,.$ (12)

As customary we assume neutrinos are ultrarelativistic, so the baseline length
for neutrino oscillation is $L\approx t$. In the limit $m_{i}\ll p_{i}$,
$i=1,2,3$, and utilizing the same-momentum approximation, $p_{1}\approx
p_{2}\approx p_{3}\approx p\approx E$, the oscillation phase is

$\Delta_{ij}\equiv\frac{E_{i}-E_{j}}{2}=\frac{L\Delta m_{ij}^{2}}{4E}\,,$ (13)

where $\Delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$ are the neutrino mass
squared differences. It is straightforward to show that in the two neutrino
approximation with mixing angle $\theta$ and oscillation frequency
$\Delta=L\Delta m_{ij}^{2}/4E$ the oscillation probability reduces to

$P_{\ell\ell^{\prime}}=\sin^{2}(2\theta)\sin^{2}\Delta,\quad\ell\neq\ell^{\prime}\,.$
(14)

### Neutrino coherence length

The standard derivation of the neutrino oscillation transition probability
follows from solving a purely quantum mechanical Schrödinger equation with a
plane wave ansatz. At the leading order this produces the correct expression,
but sub-leading effects should also be taken into account. Contrary to the
plane wave approximation, neutrino production and detection occur in a finite,
localized, spacetime region and consequently are more properly modelled in
terms of wave packets which posses uncertain energy and momentum. As the
neutrino mass eigenstates are non-degenerate, the wave packets will separate
over time and the oscillation of the transition probability will be
exponentially suppressed by a characteristic coherence length $L_{\text{coh}}$
at which the wave packet separation starts to be significant. After a baseline
of a few coherence lengths, the transition probabilities can be regarded as
constant, signalling that decoherence took place. The expression for the
transition probability computed in the wave packet formalism accounting for
the effect is given by Giunti and Kim (2007)

$P_{\ell\ell^{\prime}}=\sum\limits_{i,j=1}^{3}U_{\ell
i}^{*}U_{\ell^{\prime}i}U_{\ell
j}U_{\ell^{\prime}j}^{*}\exp\left(-2i\Delta_{ij}-\left(\frac{L}{L_{ij}^{\text{coh}}}\right)^{2}\right)\,,$
(15)

where the coherence length is

$L^{\text{coh}}_{ij}=\frac{4\sqrt{2}E_{\nu}\Delta x}{\Delta m_{ij}^{2}}\,,$
(16)

and $\Delta x$ is the neutrino position uncertainty.

### Nonstandard neutrino interactions

In a medium with electron number density $N_{e}$, the Hamiltonian regulating
neutrino oscillations is given by

$H=\frac{1}{2E}U^{\dagger}\begin{pmatrix}0&0&0\\\ 0&\Delta m_{21}^{2}&0\\\
0&0&\Delta m_{31}^{2}\end{pmatrix}U+\sqrt{2}G_{F}N_{e}\begin{pmatrix}1&0&0\\\
0&0&0\\\ 0&0&0\end{pmatrix}\,,$ (17)

where $G_{F}$ is the Fermi constant. The neutrino mixing matrices appearing in
the above equation rotate the matrix hosting the squared mass differences to
the flavour basis where interactions are diagonal. Theories beyond the SM,
including Type II seesaw Minkowski (1977); Yanagida (1979); Gell-Mann  _et
al._ (1979); Mohapatra and Senjanovic (1980); Schechter and Valle (1980), the
Zee-Babu model Zee (1980); Babu (1988) and $U(1)$ extensions coupling to
neutrino sector Péli and Trócsányi (2022); Trócsányi (2020); Iwamoto  _et al._
(2022, 2021); Kärkkäinen and Trócsányi (2022, 2023), can modify neutrino
oscillations upon the introduction of nonstandard neutrino interactions (NSI).
These are described with dimension-6 effective operators in the Lagrangian:

$\mathcal{L}_{\text{NSI}}=-2\sqrt{2}G_{F}\varepsilon^{ff^{\prime},C}_{\ell\ell^{\prime}}(\overline{\nu_{L,\alpha}}\gamma^{\mu}\nu_{L,\beta})(\overline{f}\gamma^{\mu}P_{C}f^{\prime})\,,$
(18)

where a sum over chiralities ($C=L,R$), fermions $(f,f^{\prime})$ and lepton
flavours $(\ell,\ell^{\prime}=e,\mu,\tau)$ is implied. The NSI parameters
$\varepsilon^{ff^{\prime},C}_{\ell\ell^{\prime}}$ are dimensionless and in
general complex numbers. Possible non-vanishing imaginary parts induce CP
violation (CPV) in neutrino sector even if the standard CP phase $\delta$
vanishes. The absolute values of the NSI parameters can be interpreted as the
strength of the interaction with respect to weak interaction.

Integrating out the background fermion fields, the contribution of NSI is
generally modeled in a Hamiltonian

$\displaystyle H_{\text{NSI}}$
$\displaystyle=\sum\limits_{f}\sqrt{2}G_{F}N_{f}\begin{pmatrix}\varepsilon^{f}_{ee}&\varepsilon^{f}_{e\mu}&\varepsilon^{f}_{e\tau}\\\
\varepsilon^{f}_{\mu e}&\varepsilon^{f}_{\mu\mu}&\varepsilon^{f}_{\mu\tau}\\\
\varepsilon^{f}_{\tau
e}&\varepsilon^{f}_{\tau\mu}&\varepsilon^{f}_{\tau\tau}\\\ \end{pmatrix}\,,$
(19)

where $N_{f}$ is the number density of the fermion $f$ and
$\varepsilon_{\ell\ell^{\prime}}^{f}=\varepsilon^{f,L}_{\ell\ell^{\prime}}+\varepsilon^{f,R}_{\ell\ell^{\prime}}$.
Note that hermiticity requires
$\varepsilon_{\ell\ell^{\prime}}^{f,*}=\varepsilon_{\ell^{\prime}\ell}^{f}$.
The inclusion of this operator induces subleading effects in neutrino flavour
transitions that modify the standard oscillation framework by changing the
oscillation probability $P_{\ell\ell^{\prime}}$ determined by the composition
of the medium. In the following we will derive relations between NSI and GUP
parameters and use the current experimental bounds on the former to constrain
the latter.

###  III.2 GUP-I

Having summarized the standard results for neutrino oscillation probabilities,
coherence length and NSI, we proceed to detail the how considered GUPs
scenario modify these quantities starting with the GUP-I model in Eq. (II).

#### Neutrino oscillations

Assuming that $m_{i}\ll p$, $i=1,2,3$, we can expand the dispersion relation
in Eq. (5) as

$\displaystyle E$ $\displaystyle\simeq p\sqrt{1-2\beta
p^{2}}+\frac{m^{2}}{2p\sqrt{1-2\beta p^{2}}}+\mathcal{O}(m^{4})\,,$ (20)

recovering the usual expression in the limit $\beta\rightarrow 0$. Denoting
the oscillation frequency in the standard case ($\beta=0$) as $\Delta^{(0)}$,
the GUP-I scenario gives

$\Delta^{(I)}_{ij}=\frac{\Delta^{(0)}_{ij}}{\sqrt{1-2\beta E^{2}}}\,,$ (21)

and the oscillation frequency is consequently increased as the square root is
necessarily smaller than unity.

Consider a case where one oscillation channel is dominant. The related
transition probability is given by $(\ell\neq\ell^{\prime})$

$P_{\ell\ell^{\prime}}=\sin^{2}(2\theta)\sin^{2}\left(\frac{\Delta^{(0)}}{\sqrt{1-2\beta
E^{2}}}\right)\,,$ (22)

and assuming $2\beta E^{2}\ll 1$ it gives

$\displaystyle P_{\ell\ell^{\prime}}$
$\displaystyle\simeq\sin^{2}(2\theta)(\sin^{2}\Delta^{(0)}+2\Delta^{(0)}\beta
E^{2}\cos\Delta^{(0)})\,.$ (23)

Therefore, the difference in oscillation probability induced by the GUP is

$\Delta P=2\Delta^{(0)}\beta E^{2}\sin^{2}(2\theta)\cos\Delta^{(0)}\sim\beta
E^{2}\,.$ (24)

The effect is illustrated in Fig. 1, where we plot the change in the
$P_{e\mu}$ and $P_{e\tau}$ transitions as a function of the experiment
baseline for different values of the GUP parameter and considering a typical
neutrino beam energy of 2 GeV.

The current error resolution on transition probabilities, $\Delta P\sim
10^{-2}$, can be used to estimate the maximum GUP effect allowed by current
experiments, thereby yielding the exclusion limit indicated by the gray area.
The corresponding bound is

$\beta
E^{2}=\frac{\beta}{\text{GeV}^{-2}}\times\left(\frac{E}{\text{GeV}}\right)^{2}\lesssim
10^{-2}\,,$ (25)

and implies the following constraint on the dimensionless GUP-I parameter:

$\beta_{0}\lesssim 10^{36}\times\left(\frac{\text{GeV}}{E}\right)^{2}\,.$ (26)

A similar result holds true for the experimental constraints cast by
measurements of disappearance oscillation probabilities, as shown from the
plot in Fig. 2 for a value $\beta=0.01$ GeV-2 of the GUP parameter.

#### Nonstandard interactions

Retaining the first nontrivial order of approximation in the dispersion
relation (20),

$E\simeq p\sqrt{1-2\beta p^{2}}+\frac{m^{2}}{2p}(1-2\beta p^{2})^{-1/2},$ (27)

we can write the Hamiltonian for neutrino oscillations:

$H=\frac{1}{2E}U^{\dagger}\begin{pmatrix}0&0&0\\\ 0&\Delta m_{21}^{2}&0\\\
0&0&\Delta m_{31}^{2}\end{pmatrix}U(1+\beta E^{2})\,,$ (28)

where the perturbation $\beta E^{2}\ll 1$ is the NSI term due to GUP. By
solving via matching for $\beta$ and taking the PMNS matrix elements to be
$\mathcal{O}(1)$, we find

$\beta=\frac{2\sqrt{2}G_{F}N_{e}\varepsilon}{E\Delta m_{ij}^{2}}\,,$ (29)

approximately yielding for the magnitude of $\beta$

$|\beta|\sim\frac{N_{e}}{N_{A}/\text{cm}^{3}}\times\frac{10^{-5}\text{
eV}^{2}}{\Delta
m_{ij}^{2}}\times\frac{|\varepsilon|}{10^{-2}}\frac{\text{GeV}}{E}\times
10^{-19}\text{ eV}^{-2}\,,$ (30)

or, in terms of the dimensionless quantity $\beta_{0}$,

$\beta_{0}\lesssim 10^{37}\times\frac{\text{GeV}}{E}\,.$ (31)

Although experiments utilizing lower energy neutrinos seem more suitable for
constraining $\beta_{0}$ through the related NSI, the present experimental
sensitivities are far from casting stringent bounds. For instance, the
COHERENT Akimov  _et al._ (2017, 2021); Giunti (2020); Kärkkäinen and
Trócsányi (2023) and BOREXINO Agarwalla  _et al._ (2020) experiments yield
$|\varepsilon|\lesssim 10$ and essentially result in no bound on the GUP
parameter of interest. However, the future experiments Hyper-K Abe  _et al._
(2018) and JUNO Abusleme  _et al._ (2022) are expected to probe neutrino
oscillation parameters related to the solar flux—with neutrino energies of a
few MeV— reaching a combined sensitivity to NSI that would constrain
$|\varepsilon|\lesssim 0.1$, corresponding to $\beta_{0}\lesssim 10^{35}$.

#### Coherence length

GUPs also modify the neutrino coherence length as it is directly sensitive to
the uncertainty in the neutrino position. For the GUP-I scenario, considering
terms beyond the plane wave approximation, we have

$L^{\text{coh,(I)}}_{ij}=\frac{L_{ij}^{\text{coh,(0)}}}{1+3\beta
E_{\nu}^{2}}\,,$ (32)

where $\Delta x$ is the neutrino position uncertainty and
$L_{ij}^{\text{coh,(0)}}$ the standard coherence length in Eq. (16). The
latter is clearly recovered in the limit $\beta\rightarrow 0$.

We find that the difference in transition probability induced by GUP through
the change in the coherence length is given by

$\Delta P\sim\beta E^{2}\left(\frac{L}{L^{\text{coh},(0)}}\right)^{2}\,,$ (33)

in the limit $\beta E^{2}\ll 1$ and $L\ll L^{\text{coh}^{(0)}}$. Although the
effect is subdominant with respect to the contribution in Eq. (24), we include
it in the computation of the transition probability differences shown in Fig.
3. In the plot, we set baseline and neutrino beam energy to the values proper
of the DUNE experiment and analyze the power of the latter to constrain the
dimensionless parameter $\beta_{0}$. As we can see, the bound obtained (gray
exclusion area) is in line with the results obtained for neutrino oscillations
in the plane wave limit, confirming that changes in the coherence length yield
subdominant effects at this baselines. The green bands indicate the exclusion
bounds obtainable with the foreseen sensitivity of the DUNE experiment Abi
_et al._ (2020).

###  III.3 GUP-II

We now repeat the previous analysis for the second GUP scenario considered,
specified in Eq. (7).

#### Neutrino oscillations

Expanding, in this case, the dispersion relation (8) in the limit $\alpha p\ll
1$ we obtain

$E=p|1-\alpha p|+\frac{m^{2}}{2p}(1+\alpha p)+\mathcal{O}(m^{4})\,,$ (34)

corresponding to an oscillation frequency for the GUP-II case of

$\Delta_{ij}^{(II)}=\frac{\Delta_{ij}^{(0)}}{|1-\alpha E|}\,.$ (35)

Similarly to GUP-I case, in two flavour approximation the transition
probability for $\nu_{\ell}\rightarrow\nu_{\ell^{\prime}}$ oscillation is
given by

$P_{\ell\ell^{\prime}}=\sin^{2}(2\theta)\sin^{2}\left(\frac{\Delta^{(0)}}{|1-\alpha
E|}\right)\,.$ (36)

With the further assumption $\alpha E\ll 1$, the difference in oscillation
probabilities between the standard and the GUP-II relations can be written as

$\Delta P=2\Delta^{(0)}\alpha\sin^{2}(2\theta)E\cos\Delta^{(0)}\sim\alpha
E\,.$ (37)

The current experimental resolution proper of transition probability
measurements, $\Delta P\sim 10^{-2}$, implies

$\alpha E=\frac{\alpha}{\text{GeV}^{-1}}\times\frac{E}{\text{GeV}}\lesssim
10^{-2}\,.$ (38)

Extracting the typical scale that appears in the definition of $\alpha$,
$\alpha\simeq 8\alpha_{0}\times 10^{-20}\;\mathrm{GeV}^{-1}$ the above
equation can be recast as a constraint for the dimensionless variable
$\alpha_{0}$:

$\alpha_{0}\lesssim 10^{17}\times\frac{\text{GeV}}{E}\,.$ (39)

#### Nonstandard interaction

Retaining again only the first terms in the dispersion relation expansion

$E=p|1-\alpha p|+\frac{m^{2}}{2p}(1+\alpha p)\,,$ (40)

we proceed as in the GUP-I case using the same-momentum approximation and
disregarding terms proportional to the unit matrix. The relevant Hamiltonian
describing neutrino oscillations is then given by

$H=\frac{1}{2E}U^{\dagger}\begin{pmatrix}0&0&0\\\ 0&\Delta m_{21}^{2}&0\\\
0&0&\Delta m_{31}^{2}\end{pmatrix}U(1+\alpha E)\,,$ (41)

in which the extra term proportional to $\alpha$ can be interpreted as a
matter NSI term. Via matching, we find

$\alpha=\frac{2\sqrt{2}G_{F}N_{e}\varepsilon}{\Delta m_{ij}^{2}}\,,$ (42)

where we take the PMNS matrix elements to be $\mathcal{O}(1)$. We notice that
in the context of NSI

$\alpha=\beta E\,,$ (43)

and since $\beta\sim E^{-1}$, the expression for the GUP-II parameter $\alpha$
does not involve the neutrino energy. We estimate

$|\alpha|\sim\frac{N_{e}}{N_{A}/\text{cm}^{3}}\times\frac{10^{-5}\text{
eV}^{2}}{\Delta m_{ij}^{2}}\times\frac{|\varepsilon|}{10^{-2}}\times
10^{-10}\text{ eV}^{-1}\,,$ (44)

and obtain the following order-of-magnitude approximation for the current
experimental constraint on the GUP-II parameter $|\alpha|\lesssim 10^{-10}$
eV-1 or, equivalently:

$\alpha_{0}\lesssim 10^{18}\,.$ (45)

Figure 1: Difference between the transition probabilities obtained for the
standard case and in presence of the GUP-I scenario, as a function of the
neutrino baseline $L$ and for $E_{\nu}=2$ GeV. The dashed vertical lines
indicate the DUNE experiment baseline Abi  _et al._ (2020) whereas the gray
areas are the exclusion contours determined by the present experimental
sensitivity. Top panel: $|\Delta P(\nu_{e}\rightarrow\nu_{\mu})|$. Bottom
panel: $|\Delta P(\nu_{e}\rightarrow\nu_{\tau})|$.  Figure 2: The difference
in neutrino oscillation transition probabilities computed for $\beta=0.01$
GeV-2 ($\beta_{0}\sim 10^{36}$) and the standard case. For the computation we
have assumed normal mass hierarchy, upper $\theta_{23}$ octant, no CPV and set
the neutrino energy to $E_{\nu}=2$ GeV, which is the expected beam energy of
the DUNE experiment with baseline $L=1300$ km Abi  _et al._ (2020). We have
set the neutrino energy spread to $\sigma_{E}/E=0.02.$ Figure 3: Differences
between the standard transition probabilities and the corresponding quantities
obtained in the GUP-I scenario as a function of the dimensionless parameter
$\beta_{0}$. The plot uses the DUNE experiment baseline and neutrino energy.
The gray areas are the exclusion contours determined by the present
experimental sensitivity, the green bands indicate instead the bounds
obtainable with the expected DUNE sensitivity.

#### Coherence length

The coherence length resulting from the GUP-II scenario is

$L_{ij}^{\text{coh,(II)}}=\frac{L_{ij}^{\text{coh,(0)}}}{1+2\alpha
E_{\nu}}\,,$ (46)

and correctly recovers the standard case for $\alpha\rightarrow 0$.

Similar to the previous analysis, we find that the difference in transition
probability is then given by

$\Delta P\sim\alpha E\left(\frac{L}{L^{\text{coh},(0)}}\right)^{2}\,,$ (47)

in the limit $\alpha E\ll 1$ and $L\ll L^{\text{coh},(0)}$. As we can see,
also for GUP-II scenario the effect is subdominant with respect to that in Eq.
(37) at current experiments.

##  IV Results

In this letter we have investigated the experimental feasibility of using the
low energy neutrino data to probe the phenomenological effects of two GUP
scenarios. Whereas experiments are not sensible enough to distinguish between
the two cases analyzed, we find that the data can indeed constrain the
involved GUP parameters. In Tab. 1 we collect the constraints obtained from
the results in Secs. III.2 and III.3 considering the specifics of three
upcoming neutrino experiments: Hyper-Kamiokande Abe  _et al._ (2018), DUNE Abi
_et al._ (2020) and JUNO Abusleme  _et al._ (2022). The quoted bounds are
mainly driven by the modifications that the considered GUP scenario yield on
the neutrino oscillation pattern computed in the plane wave approximation.
These constraints are comparable to the—marginally milder—exclusions obtained
by modeling the GUP effects as nonstandard neutrino interactions and strongly
outclass the limit obtained from the neutrino coherence length in current
experimental setups.

Experiment |  $L$/km |  $E_{\nu}^{\text{peak}}$/MeV | $\Delta P$ |  | $\alpha_{0}$ |  | $\beta_{0}$
---|---|---|---|---|---|---|---
DUNE Abi  _et al._ (2020) | 1300 | 2000 |  $0.5$ % |  | $10^{16}$ |  | $10^{36}$
Hyper-K Abe  _et al._ (2018) | 295 | 600 |  $1$ % |  | $10^{18}$ |  | $10^{38}$
JUNO Abusleme  _et al._ (2022) | 53 | 4.5 |  $1$ % |  | $10^{18}$ |  | $10^{41}$
Table 1: The future neutrino oscillation experiments, their baselines, the
most probable neutrino energies and foreseen measurement accuracy. The last
two columns shown the achievable upper bounds on the GUP-parameters
$\alpha_{0}$ and $\beta_{0}$.

To put our results into perspective, we refer the reader to the recent review
paper Bosso  _et al._ (2023b) and the tables contained, which list the upper
bounds on the parameter $\beta_{0}$ derived from various gravitational and
non-gravitational experiments and observations. As we can see, the considered
future neutrino oscillation experiments Abe  _et al._ (2018); Abi  _et al._
(2020); Abusleme  _et al._ (2022) could impose bounds stricter than most of
the ones presented in Ref. Bosso  _et al._ (2023b).

###### Acknowledgements.

This work was supported by the Estonian Research Council grants MOBJD1202,
PRG803, RVTT3, RVTT7, and by the CoE program TK202 “Fundamental Universe”.

## References

  * Amati  _et al._ (1987) D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 197, 81 (1987).
  * Konishi  _et al._ (1990) K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234, 276 (1990).
  * Rovelli and Smolin (1995) C. Rovelli and L. Smolin, Nucl. Phys. B 442, 593 (1995), [Erratum: Nucl.Phys.B 456, 753–754 (1995)], arXiv:gr-qc/9411005 .
  * Arkani-Hamed  _et al._ (1998) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 429, 263 (1998), arXiv:hep-ph/9803315 .
  * Antoniadis  _et al._ (1998) I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 436, 257 (1998), arXiv:hep-ph/9804398 .
  * Arkani-Hamed  _et al._ (1999) N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Rev. D 59, 086004 (1999), arXiv:hep-ph/9807344 .
  * Maggiore (1993a) M. Maggiore, Phys. Lett. B 304, 65 (1993a), arXiv:hep-th/9301067 .
  * Garay (1995) L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995), arXiv:gr-qc/9403008 .
  * Kempf  _et al._ (1995) A. Kempf, G. Mangano, and R. B. Mann, Phys. Rev. D 52, 1108 (1995), arXiv:hep-th/9412167 .
  * Adler and Santiago (1999) R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14, 1371 (1999), arXiv:gr-qc/9904026 .
  * Chen and Adler (2003) P. Chen and R. J. Adler, Nucl. Phys. B Proc. Suppl. 124, 103 (2003), arXiv:gr-qc/0205106 .
  * Das and Vagenas (2008) S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008), arXiv:0810.5333 [hep-th] .
  * Adler (2010) R. J. Adler, Am. J. Phys. 78, 925 (2010), arXiv:1001.1205 [gr-qc] .
  * Maggiore (1994) M. Maggiore, Phys. Rev. D 49, 5182 (1994), arXiv:hep-th/9305163 .
  * Maggiore (1993b) M. Maggiore, Phys. Lett. B 319, 83 (1993b), arXiv:hep-th/9309034 .
  * Scardigli (1999) F. Scardigli, Phys. Lett. B 452, 39 (1999), arXiv:hep-th/9904025 .
  * Adler  _et al._ (2001) R. J. Adler, P. Chen, and D. I. Santiago, Gen. Rel. Grav. 33, 2101 (2001), arXiv:gr-qc/0106080 .
  * Myung  _et al._ (2007) Y. S. Myung, Y.-W. Kim, and Y.-J. Park, Phys. Lett. B 645, 393 (2007), arXiv:gr-qc/0609031 .
  * Nozari and Hamid Mehdipour (2008) K. Nozari and S. Hamid Mehdipour, EPL 84, 20008 (2008), arXiv:0804.4221 [gr-qc] .
  * Bargueño and Vagenas (2015) P. Bargueño and E. C. Vagenas, Phys. Lett. B 742, 15 (2015), arXiv:1501.03256 [hep-th] .
  * Carr  _et al._ (2015) B. J. Carr, J. Mureika, and P. Nicolini, JHEP 07, 052 (2015), arXiv:1504.07637 [gr-qc] .
  * Casadio  _et al._ (2018) R. Casadio, P. Nicolini, and R. da Rocha, Class. Quant. Grav. 35, 185001 (2018), arXiv:1709.09704 [hep-th] .
  * Alonso-Serrano  _et al._ (2018) A. Alonso-Serrano, M. P. Dabrowski, and H. Gohar, Phys. Rev. D 97, 044029 (2018), arXiv:1801.09660 [gr-qc] .
  * Ong (2018) Y. C. Ong, JCAP 09, 015 (2018), arXiv:1804.05176 [gr-qc] .
  * Gangopadhyay and Dutta (2018) S. Gangopadhyay and A. Dutta, Adv. High Energy Phys. 2018, 7450607 (2018), arXiv:1805.11962 [physics.gen-ph] .
  * Mureika (2019) J. R. Mureika, Phys. Lett. B 789, 88 (2019), arXiv:1812.01999 [gr-qc] .
  * Knipfer  _et al._ (2019) M. Knipfer, S. Köppel, J. Mureika, and P. Nicolini, JCAP 08, 008 (2019), arXiv:1905.03233 [gr-qc] .
  * Carr  _et al._ (2020) B. Carr, H. Mentzer, J. Mureika, and P. Nicolini, Eur. Phys. J. C 80, 1166 (2020), arXiv:2006.04892 [gr-qc] .
  * Zhou  _et al._ (2022) X. Zhou, Z.-W. Feng, and S.-Q. Zhou, Front. in Phys. 10, 887410 (2022), arXiv:2008.01661 [gr-qc] .
  * Carr (2022) B. J. Carr, Front. Astron. Space Sci. 9, 1008221 (2022), arXiv:2302.12609 [gr-qc] .
  * Feng  _et al._ (2017) Z.-W. Feng, S.-Z. Yang, H.-L. Li, and X.-T. Zu, Phys. Lett. B 768, 81 (2017), arXiv:1610.08549 [hep-ph] .
  * Bhattacharyya  _et al._ (2020) S. Bhattacharyya, S. Gangopadhyay, and A. Saha, Class. Quant. Grav. 37, 195006 (2020), arXiv:2005.09454 [gr-qc] .
  * Diab and Tawfik (2022) A. M. Diab and A. N. Tawfik, Adv. High Energy Phys. 2022, 9351511 (2022), arXiv:2005.03999 [physics.gen-ph] .
  * Das  _et al._ (2021) A. Das, S. Das, N. R. Mansour, and E. C. Vagenas, Phys. Lett. B 819, 136429 (2021), arXiv:2101.03746 [gr-qc] .
  * Das  _et al._ (2022) S. Das, S. Shankaranarayanan, and V. Todorinov, Phys. Lett. B 835, 137511 (2022), arXiv:2208.11095 [gr-qc] .
  * Benczik  _et al._ (2005) S. Benczik, L. N. Chang, D. Minic, and T. Takeuchi, Phys. Rev. A 72, 012104 (2005), arXiv:hep-th/0502222 .
  * Bambi and Urban (2008) C. Bambi and F. R. Urban, Class. Quant. Grav. 25, 095006 (2008), arXiv:0709.1965 [gr-qc] .
  * Marin  _et al._ (2013) F. Marin _et al._ , Nature Phys. 9, 71 (2013).
  * Scardigli and Casadio (2015) F. Scardigli and R. Casadio, Eur. Phys. J. C 75, 425 (2015), arXiv:1407.0113 [hep-th] .
  * Bernaldez  _et al._ (2023) J. P. R. Bernaldez, A. G. Abac, and R. E. S. Otadoy, Annals Phys. 457, 169402 (2023), arXiv:2210.09876 [hep-ph] .
  * Artigas  _et al._ (2024) D. Artigas, K. Martineau, and J. Mielczarek, Phys. Rev. D 109, 024028 (2024), arXiv:2308.13788 [quant-ph] .
  * Pachol and Wojnar (2023) A. Pachol and A. Wojnar, Eur. Phys. J. C 83, 1097 (2023), arXiv:2307.03520 [gr-qc] .
  * Kozak  _et al._ (2023) A. Kozak, A. Pachol, and A. Wojnar, (2023), arXiv:2310.00913 [gr-qc] .
  * Bosso  _et al._ (2023a) P. Bosso, O. Obregón, S. Rastgoo, and W. Yupanqui, (2023a), arXiv:2310.04600 [gr-qc] .
  * Wojnar (2024) A. Wojnar, Phys. Rev. D 109, 024011 (2024), arXiv:2311.14066 [gr-qc] .
  * Segreto and Montani (2024) S. Segreto and G. Montani, (2024), arXiv:2401.17113 [gr-qc] .
  * Hong  _et al._ (2024) S.-T. Hong, Y.-W. Kim, and Y.-J. Park, (2024), arXiv:2403.13608 [gr-qc] .
  * Ali and Wojnar (2024) A. F. Ali and A. Wojnar, Class. Quant. Grav. 41, 105001 (2024), arXiv:2401.05941 [gr-qc] .
  * Scardigli  _et al._ (2017) F. Scardigli, G. Lambiase, and E. Vagenas, Phys. Lett. B 767, 242 (2017), arXiv:1611.01469 [hep-th] .
  * Bosso  _et al._ (2023b) P. Bosso, G. G. Luciano, L. Petruzziello, and F. Wagner, Class. Quant. Grav. 40, 195014 (2023b), arXiv:2305.16193 [gr-qc] .
  * Sprenger  _et al._ (2011) M. Sprenger, M. Bleicher, and P. Nicolini, Class. Quant. Grav. 28, 235019 (2011), arXiv:1011.5225 [hep-ph] .
  * Torri and Miramonti (2024) M. D. C. Torri and L. Miramonti, (2024), arXiv:2404.04076 [gr-qc] .
  * Ali  _et al._ (2009) A. F. Ali, S. Das, and E. C. Vagenas, Phys. Lett. B 678, 497 (2009), arXiv:0906.5396 [hep-th] .
  * Das  _et al._ (2010) S. Das, E. C. Vagenas, and A. F. Ali, Phys. Lett. B 690, 407 (2010), [Erratum: Phys.Lett.B 692, 342–342 (2010)], arXiv:1005.3368 [hep-th] .
  * Esteban  _et al._ (2020) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, JHEP 09, 178 (2020), arXiv:2007.14792 [hep-ph] .
  * Giunti and Kim (2007) C. Giunti and C. W. Kim,  _Fundamentals of Neutrino Physics and Astrophysics_ (2007).
  * Minkowski (1977) P. Minkowski, Phys. Lett. B 67, 421 (1977).
  * Yanagida (1979) T. Yanagida, Conf. Proc. C 7902131, 95 (1979).
  * Gell-Mann  _et al._ (1979) M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. C 790927, 315 (1979), arXiv:1306.4669 [hep-th] .
  * Mohapatra and Senjanovic (1980) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
  * Schechter and Valle (1980) J. Schechter and J. W. F. Valle, Phys. Rev. D 22, 2227 (1980).
  * Zee (1980) A. Zee, Phys. Lett. B 93, 389 (1980), [Erratum: Phys.Lett.B 95, 461 (1980)].
  * Babu (1988) K. S. Babu, Phys. Lett. B 203, 132 (1988).
  * Péli and Trócsányi (2022) Z. Péli and Z. Trócsányi, Phys. Rev. D 106, 055045 (2022), arXiv:2204.07100 [hep-ph] .
  * Trócsányi (2020) Z. Trócsányi, Symmetry 12, 107 (2020), arXiv:1812.11189 [hep-ph] .
  * Iwamoto  _et al._ (2022) S. Iwamoto, K. Seller, and Z. Trócsányi, JCAP 01, 035 (2022), arXiv:2104.11248 [hep-ph] .
  * Iwamoto  _et al._ (2021) S. Iwamoto, T. J. Kärkkäinen, Z. Péli, and Z. Trócsányi, Phys. Rev. D 104, 055042 (2021), arXiv:2104.14571 [hep-ph] .
  * Kärkkäinen and Trócsányi (2022) T. J. Kärkkäinen and Z. Trócsányi, J. Phys. G 49, 045004 (2022), arXiv:2105.13360 [hep-ph] .
  * Kärkkäinen and Trócsányi (2023) T. J. Kärkkäinen and Z. Trócsányi, Phys. Rev. D 107, 115020 (2023), arXiv:2301.06621 [hep-ph] .
  * Akimov  _et al._ (2017) D. Akimov _et al._ (COHERENT), Science 357, 1123 (2017), arXiv:1708.01294 [nucl-ex] .
  * Akimov  _et al._ (2021) D. Akimov _et al._ (COHERENT), Phys. Rev. Lett. 126, 012002 (2021), arXiv:2003.10630 [nucl-ex] .
  * Giunti (2020) C. Giunti, Phys. Rev. D 101, 035039 (2020), arXiv:1909.00466 [hep-ph] .
  * Agarwalla  _et al._ (2020) S. K. Agarwalla _et al._ (Borexino), JHEP 02, 038 (2020), arXiv:1905.03512 [hep-ph] .
  * Abe  _et al._ (2018) K. Abe _et al._ (Hyper-Kamiokande), (2018), arXiv:1805.04163 [physics.ins-det] .
  * Abusleme  _et al._ (2022) A. Abusleme _et al._ (JUNO), Prog. Part. Nucl. Phys. 123, 103927 (2022), arXiv:2104.02565 [hep-ex] .
  * Abi  _et al._ (2020) B. Abi _et al._ (DUNE), Eur. Phys. J. C 80, 978 (2020), arXiv:2006.16043 [hep-ex] .


#
Change in the orbital period of a binary system due to an outburst in a windy
accretion disc

Author Affiliation Country Contact Details Affiliation Country Contact Details

###### Abstract

We consider a new mechanism for the removal of the angular momentum from an
X-ray binary system and the change in its orbital period — the mass loss in
the form of a wind from an accretion disc. Both observations and models
predict powerful winds from discs in X-ray transients. We have obtained an
analytical estimate for the increase in the orbital period of a binary system
with a wind from the disc during an outburst, and quantitative estimates are
given for the systems XTE J1118$+$480, A0620$-$00 and GRS 1124$-$68\.
Resulting rate of period grow is of order of the observed rates of secular
decrease in the period. We also compare the predicted rates of change in the
period of a binary system due to the flow of matter into the disc and outflow
from the second Lagrange point with the observations. It is concluded that the
above mechanisms cannot explain the observed secular decrease in the period of
the three X-ray Novae, and it is necessary to consider a circumbinary disc
that drains the binary’s angular momentum.

#### Keywords:

accretion, accretion discs – X-rays: binaries – X-rays: bursts – stars:
individual: stars: individual: XTE J1118$+$480– stars: individual: A0620$-$00
– stars: individual: GRS 1124$-$68

##  1 Introduction

Orbital period variations provide important information about the parameters
of a close binary system, as well as the processes taking place inside it: the
flow of matter between the components, the evolution of a companion star, etc.
In close binaries, the change in the orbital period is mainly associated with
a decrease in the angular momentum (Tutukov & Yungelson, 1971, Cherepaschuk,
2013). This decrease in the angular momentum of a close binary system can be
caused by various mechanisms, among which there are three principal ones: mass
loss by the system (Rappaport et al., 1982), magnetic braking (Verbunt &
Zwaan, 1981, Tutukov, 1983), and gravitational waves (Landau & Lifshitz,
1962).

In particular, the period changes are observed in low-mass X-ray binaries
(LMXB). In these systems, matter flows from the optical star to the compact
component, accompanied by the formation of a hot accretion disc, which
produces X-ray radiation.

González Hernández et al. (2012) (hereinafter GH12) were able first to detect
a change in a LMXB orbital period when analyzing the observations of the XTE
J1118$+$480 system. It turned out that the orbital period decreased in the
system. In later works (González Hernández et al., 2014, González Hernández et
al., 2017, hereinafter GH14 and GH17), a decrease in the period was also found
for A0620$-$00 and GRS 1124$-$68\. The authors believe that the main cause of
the rapid decrease in the period in these LMXB is magnetic braking; however,
the model values that they have obtained, even under an assumption of a
sufficiently strong magnetic field, are one or two orders of magnitude lower
than the observed ones.

In this paper, we consider another mechanism for the loss of angular momentum
from a binary system — the mass loss by the system. LMXBs exhibit repeated
outbursts caused by instabilities in the disc or unsteady flow of matter
between the components (e.g., Lasota, 2016, Bath et al., 1974, Bath, 1975,
Bath & Pringle, 1981, Hameury, 2020). During an outburst, the accretion rate
on the compact component rises by several orders of magnitude, and most
radiation from a LMXB goes in the X-ray range (see, for example Chen et al.,
1997). Presumably, during an outburst there is an outflow in the form of a
wind from the accretion disc around the compact object.

The presence of such a wind in the LMXBs is supported by modern observations
indicating the expansion of ionized matter. Outflows from accretion discs are
observed both as X-ray absorption lines shifted to the blue part of the
spectrum (e.g., Ueda et al., 2009, Díaz Trigo & Boirin, 2016), and as line
broadening in the optical range (e.g., Muñoz-Darias et al., 2019, Casares et
al., 2019, Charles et al., 2019).

In most cases, X-ray absorption lines of highly ionized Fe are observed in
systems with the inclination of more than $50^{\circ}$. Therefore, the
absorbing plasma should have a higher density closer to the disc, which
suggests that it is a substance flowing out of the disc (Higginbottom & Proga,
2015).

Observational rates of the mass loss in the wind are obtained for the
intermediate mass X-ray binary (IMXB) Her X-1 in the work of Kosec et al.
(2020) and the estimates strongly depend on the wind geometry. According to
the paper, if the wind moves along the disc then the mass loss rate in the
wind in Her X-1 is approximately equal to the accretion rate onto the compact
object. However, if the wind is spherically symmetric, then the mass-loss
estimate increases by an order of magnitude. In the work by Ponti et al.
(2012), observational estimates of the rate of matter outflow from several
X-ray transients are presented, from which it follows that the ratios of the
rate of matter loss in the wind to the accretion rate onto a black hole are in
the range from $1$ to $10$. In hydrodynamical numerical simulations of
thermally driven disc winds (Higginbottom et al., 2017, Luketic et al., 2010,
Higginbottom et al., 2019), this ratio is obtained to lie in the range from
$2$ to $15$. These studies presume that the heating of the matter occurs due
to the irradiation of the outer parts of a disc by a central X-ray source.
Thus, both modelings and observations favour strong winds of matter from the
discs.

The problem of a change in the orbital period of a binary system as a result
of the mass loss by a component (or both) has been studied earlier. The
simplest description involves an isotropic gas release without taking into
account the effect of an ejected matter on the motion of the binary system,
and it was solved by Jeans (1928). It was assumed that the ejected matter was
removed very quickly without affecting the orbital motion. This can be
applicable in the case of strong outbursts of novae and supernovae. If the
ejection velocity is not high enough, then the ejected matter affects the
orbital motion of the binary system, not only due to the changes in its mass
and angular momentum, but also due to the gravitational effect on the system.
In addition, the ejected matter can return to the system to any of the stars,
causing additional changes in the period. For example, a viscous toroidal disc
around a binary system (circumbinary disc) may be formed, which can
effectively transport the angular momentum it receives due to tidal
interactions (Chen & Podsiadlowski, 2019, hereinafter CP19).

In addition to the wind from the accretion disc, outflow of matter from the
outer Lagrange point $L_{2}$, which is located behind the less massive
component (behind the optical star in the case of LMXB), can occur in the
system. And, unlike the wind from the disc, which is most active only during
an outburst, the flow of matter from $L_{2}$ can be constantly present and
lead to a constant decrease in the period of the system.

The article by Kruszewski (1964) provides a detailed solution to the binary
system mass-loss problem. In its formulation, it is necessary to know the
components of the velocity and the coordinates of the ejected substance, which
are not known with a sufficient accuracy. In the present paper, a simpler
problem is considered that does not take into account the effect of the
ejected matter on the orbital motion of the system as it goes to infinity.

The structure of the paper is as follows. In Section § 2 the equations of the
orbital period change model are described in detail. Namely, subsections §
2.1, § 2.2, § 2.3 are devoted to three different mechanisms: disc wind, inflow
of matter form the donor star, and outflow from Lagrange point $L_{2}$,
respectively. In section § 3 we present the results of our model and the
application to XTE J1118$+$480, A0620$-$00, and GRS 1124$-$68\. The
discussions and conclusions are provided in sections § 4 in § 5.

##  2 Model of orbital period change

###  2.1 Impact of accretion disc wind

Let the matter of the companion star flow through the Lagrange point $L_{1}$
into the Roche lobe of the compact object and become part of the accretion
disc there. Under the influence of self-irradiation, magnetic field or
radiation pressure, some of the matter can be carried away from the disc in
the form of a wind, taking with it both the mass and the angular momentum.
Thus, the total mass and total angular momentum of the binary system change.
We will assume that all the wind starts from a fixed radius in the disc.

Let’s calculate the change in the orbital period of the system. We will use
the Kepler’s third law:

$P=2\pi\,\sqrt{\frac{A^{3}}{GM}},$ (2.1)

where $M$ is the total mass in the binary system, and $P$ and $A$ are the
period and semi-major axis, respectively. Next, we find the relative change in
the orbital period of the binary $\Delta P/P$, by varying (2.1) as a function
of $A$ and $M$. As a result, we get:

$\frac{\Delta P}{P}=\frac{3}{2}\frac{\Delta A}{A}-\frac{1}{2}\frac{\Delta
M}{M},$ (2.2)

where $P$ and $\Delta P$ are the orbital period of the system and its change,
respectively. Our ultimate goal is to directly relate $\Delta M/M$ and $\Delta
P/P$. Therefore, it is necessary to express $\Delta A/A$ through $\Delta M/M$.

To calculate this change in the semi-major axis, we use the law of
conservation of angular momentum. We assume that before an outburst the disc
is a ring of mass $M_{\rm disc}$ with a characteristic radius $R_{\rm disc}$.
Further, all the ’primed’ variables (when considering the effect of the wind)
denote the state of the system after the outburst, and all the ’unprimed’
ones, the state before the outburst.

The total angular momentum of the binary system before the ejection of matter
to infinity $J$ is:

$J=J_{\rm opt}+J_{\rm x}+J_{\rm disc},$ (2.3)

where $J_{\rm opt}$, $J_{\rm x}$, and $J_{\rm disc}$ are the angular momentum
of the optical star, compact object, and accretion disc around it,
respectively. They are expressed as follows:

$\begin{gathered}J_{\rm opt}=M_{\rm opt}\,\omega_{\rm orb}\,R_{\rm
opt}^{2},\\\ J_{\rm x}=M_{\rm x}\,\omega_{\rm orb}\,R_{\rm x}^{2},\\\ J_{\rm
disc}=M_{\rm disc}(\omega_{\rm orb}\,R_{\rm x}^{2}+\sqrt{GM_{\rm x}R_{\rm
disc}}).\end{gathered}$ (2.4)

It can be seen that the angular momentum of the disc consists of the orbital
angular momentum and the angular momentum associated with the Keplerian
rotation around the relativistic star. $M_{\rm opt}$ and $M_{\rm x}$ are the
masses of the optical and compact components, $R_{\rm opt}$ and $R_{\rm x}$
are the distances from centers of mass of stars to the center of mass of the
binary system, $\omega_{\rm orb}$ is the angular velocity of orbital rotation
(see Fig. 1):

$\begin{gathered}R_{\rm opt}=A\,(M_{\rm x}+M_{\rm disc})/M,\\\ R_{\rm
x}=A\,M_{\rm opt}/M,\\\ \omega_{\rm orb}=\sqrt{\frac{GM}{A^{3}}}\,,\\\
M=M_{\rm opt}+M_{\rm x}+M_{\rm disc}\,.\end{gathered}$ (2.5)

Figure 1: Designations for a close binary system. Point C is the center of
mass of the system, $L_{1}$ and $L_{2}$ are the Lagrange points.

We assume that after the outburst the accretion disc is completely consumed:
part of the disc mass, $\Delta M_{\rm acc}$, has fallen onto the compact
object, as a result of which its mass increased to $M_{\rm
x}^{{}^{\prime}}=M_{\rm x}+\Delta M_{\rm acc}$, and the rest of the disc mass,
$\left|\Delta M_{\rm wind}\right|=M_{\rm disc}-\Delta M_{\rm acc}$ ($\Delta
M_{\rm wind}<0$), has flown out of the system as a wind. Thus, the angular
momentum of the system after the outburst:

$\begin{gathered}J^{{}^{\prime}}=J_{\rm opt}^{{}^{\prime}}+J_{\rm
x}^{{}^{\prime}}\,,\\\ J_{\rm opt}^{{}^{\prime}}=M_{\rm opt}\,\omega_{\rm
orb}^{{}^{\prime}}\,(R_{\rm opt}^{{}^{\prime}})^{2}\,,\\\ J_{\rm
x}^{{}^{\prime}}=(M_{\rm x}+\Delta M_{\rm acc})\,\omega_{\rm
orb}^{{}^{\prime}}\,(R_{\rm x}^{{}^{\prime}})^{2}\ +\Delta M_{\rm
acc}\sqrt{GM_{\rm x}R_{\rm in}},\end{gathered}$ (2.6)

where $R_{\rm in}$ is the inner radius of the disc (the radius of the last
stable orbit). We neglect the angular momentum $\Delta M_{\rm
acc}\sqrt{GM_{\rm x}R_{\rm in}}$ added to the angular momentum of a compact
object as a result of accretion, since the inner radius of the disc is $R_{\rm
in}$ is much smaller than other characteristic radii of the problem. After the
outburst, the quantities $R_{\rm opt}$, $R_{\rm x}$, $w_{\rm orb}$, and $M$
acquire new values, namely:

$\begin{gathered}R_{\rm opt}^{{}^{\prime}}=A^{{}^{\prime}}\,(M_{\rm x}+\Delta
M_{\rm acc})/M^{{}^{\prime}}\,,\\\ R_{\rm
x}^{{}^{\prime}}=A^{{}^{\prime}}\,M_{\rm opt}/M^{{}^{\prime}}\,,\\\
\omega_{\rm
orb}^{{}^{\prime}}=\sqrt{\frac{GM^{{}^{\prime}}}{(A^{{}^{\prime}})^{3}}}\\\
M^{{}^{\prime}}=M_{\rm opt}+M_{\rm x}+\Delta M_{\rm acc}\,,\end{gathered}$
(2.7)

where $A^{{}^{\prime}}$ is the value of the semi-major axis after the
outburst. The change in the angular momentum of the $J^{{}^{\prime}}-J$ system
is equal to the angular momentum carried away in the wind $\Delta J_{\rm
wind}$ ($\Delta J_{\rm wind}<0$). If the wind is launched from the radius
$R_{\rm esc}$, then it carries the angular momentum of a ring with mass
$\left|\Delta M_{\rm wind}\right|$ participating in the orbital motion and
rotation around the compact object:

$\Delta J_{\rm wind}=\Delta M_{\rm wind}(\omega_{\rm
orb}R_{x}^{2}+\sqrt{GM_{\rm x}A}\sqrt{k_{\rm esc}})<0\,,$ (2.8)

where $k_{\rm esc}\equiv R_{\rm esc}/A$. Here we take into account the
possibility that the characteristic radius of the disc changes during an
outburst: the outflow of the wind can occur from a radius larger than the
characteristic radius of the disc in a quiescent state ($R_{\rm disc}$).

By substituting (2.8) into the equation $J=J^{{}^{\prime}}-\Delta J_{\rm
wind}$ ($\Delta J_{\rm wind}<0$), we find the relation between the semi-major
axis change due to the wind influence $\Delta A_{\rm burst}\equiv
A^{{}^{\prime}}-A$ with the decrease in the total mass of the system and the
accreted mass:

$\begin{gathered}\frac{1}{2}\frac{\Delta A_{\rm burst}}{A}=\left(\frac{(M_{\rm
opt}+M_{\rm x})^{3/2}}{M_{\rm opt}\sqrt{M}_{\rm x}}\,(\sqrt{k_{\rm
esc}}-\sqrt{k_{\rm disc}})-\frac{1}{2}\right)\times\\\ \times\,\frac{\Delta
M_{\rm wind}}{M_{\rm opt}+M_{\rm x}}\,+\,\frac{(M_{\rm opt}+M_{\rm
x})^{3/2}}{M_{\rm opt}\sqrt{M}_{\rm x}}\,\sqrt{k_{\rm disc}}\,\frac{\Delta
M_{\rm acc}}{M_{\rm opt}+M_{\rm x}}\,,\end{gathered}$ (2.9)

where $k_{\rm disc}\equiv R_{\rm disc}/A$ by analogy with $k_{\rm esc}$.

If we divide expression (2.9) by the outburst duration, we can express the
average rate of change of the semi-major axis in terms of the average
accretion rate and the average rate of mass loss to the wind. Then, according
to (2.2), the period change in terms of $\left<\dot{P}_{\rm burst}\right>$,
$\left<\dot{M}_{\rm wind}\right>$, $\left<\dot{M}_{\rm acc}\right>$ and
$q=M_{\rm x}/M_{\rm opt}$ is written as follows:

$\begin{gathered}\frac{\left<\dot{P}_{\rm
burst}\right>}{P}=\frac{3}{1+q}\left(\frac{(1+q)^{3/2}}{\sqrt{q}}(\sqrt{k_{\rm
esc}}-\sqrt{k_{\rm disc}})-\frac{2}{3}\right)\times\\\
\times\,\frac{\left<\dot{M}_{\rm wind}\right>}{M_{\rm
opt}}\,+\,\frac{3\,\sqrt{1+q}}{\sqrt{q}}\sqrt{k_{\rm
disc}}\;\frac{\left<\dot{M}_{\rm acc}\right>}{M_{\rm opt}}\,.\end{gathered}$
(2.10)

Since $\left<\dot{M}_{\rm wind}\right><0$, the first term in equation (2.10)
is negative under the condition $k_{\rm esc}^{1/2}>k_{\rm
disc}^{1/2}+2/3\,(q^{1/2}/(1+q)^{3/2})$. The second term is positive and is
due to the accretion onto the compact object.

It is also necessary to take into account the following limitation following
from the physics of the process. As mentioned above, the angular momentum of
the disc $J_{\rm disc}$ consists of the orbital angular momentum $J_{\rm
disc}^{\rm orb}$ and the angular momentum associated with the Keplerian
rotation around the compact object $J_{\rm disc}^{\rm Kepl}$:

$\begin{gathered}J_{\rm disc}=J_{\rm disc}^{\rm orb}+J_{\rm disc}^{\rm
Kepl}=\\\ =M_{\rm disc}\,\omega_{\rm orb}\,R_{\rm x}^{2}+M_{\rm
disc}\sqrt{GM_{\rm x}R_{\rm disc}}.\end{gathered}$ (2.11)

The angular momentum of the disc, associated with the rotation of the matter
around the black hole, can only decrease during an outburst: the disc’s mass
decreases and the disc is decelerated by the gravitational tidal forces near
its outer boundary. As a result, the second term $J_{\rm disc}^{\rm Kepl}$
after the burst is divided between three non-negative parts: (1) angular
momentum $J_{\rm acc}^{\rm Kepl}$ of the matter falling on a compact object;
(2) angular momentum $J_{\rm wind}^{\rm Kepl}$, carried away by the wind and
(3) angular momentum $J_{P}^{\rm Kepl}$, converted into orbital motion due to
tidal forces near the outer radius of the disc:

$\begin{gathered}J_{\rm disc}^{\rm Kepl}=J_{\rm acc}^{\rm Kepl}+J_{\rm
wind}^{\rm Kepl}+J_{P}^{\rm Kepl}\,,\\\ M_{\rm disc}\sqrt{GM_{\rm x}R_{\rm
disc}}=\Delta M_{\rm acc}\sqrt{GM_{\rm x}R_{\rm in}}\,+\\\ +\,\left|\Delta
M_{\rm wind}\right|\sqrt{GM_{\rm x}R_{\rm esc}}+J_{P}^{\rm
Kepl}\,,\end{gathered}$ (2.12)

As before, the part of the angular momentum added to the angular momentum of
the compact object as a result of accretion $J_{\rm acc}^{\rm Kepl}$ is
neglected because it is small compared to $J_{\rm wind}^{\rm Kepl}$ and
$J_{\rm disc}^{\rm Kepl}$ ($R_{\rm in}\ll R_{\rm esc}$).

Accordingly, it follows from (2.12) that $\left|\Delta M_{\rm
wind}\right|\sqrt{GM_{\rm x}R_{\rm esc}}\leq M_{\rm disc}\sqrt{GM_{\rm
x}R_{\rm disc}}$. Taking this into account and introducing the parameter
$C_{\rm w}\equiv\left|\Delta M_{\rm wind}\right|/\Delta M_{\rm acc}$, we get
the restriction:

$\begin{gathered}C_{\rm w}\leq C_{\rm w}^{*}\equiv\frac{\sqrt{k_{\rm
disc}}}{\sqrt{k_{\rm esc}}-\sqrt{k_{\rm disc}}}\,.\end{gathered}$ (2.13)

It follows that for $C_{\rm w}=C_{\rm w}^{*}(q)$, that is, when the entire
initial Keplerian angular momentum of the disc $J_{\rm disc}^{\rm Kepl}$ is
carried away by the wind, the change in the orbital period of the binary is
described by the following formula (Jeans model):

$\begin{gathered}\frac{\left<\dot{P}_{\rm
burst}\right>}{P}=-\frac{2}{1+q}\,\frac{\left<\dot{M}_{\rm
wind}\right>}{M_{\rm opt}}.\end{gathered}$ (2.14)

In particular, formula (2.14) is applicable when all the ring’s matter leaves
the system without an accretion event. But such an approximation (wind without
accretion) for an X-ray nova is unjustified.

Note that the value of $C_{\rm w}^{*}(q)$ imposes a restriction on the
parameter $C_{\rm w}(q)$, that is, on the wind power, only if the launch
radius is greater than the initial radius of the disc, i.e. $k_{\rm
esc}>k_{\rm disc}$ (in this case formula (2.13) makes sense). In other cases,
the relative mass loss in the wind is not limited.

Using the introduced parameter $C_{\rm w}$,, we finally rewrite the formula
(2.10) as:

$\begin{gathered}\frac{\left<\dot{P}_{\rm
burst}\right>}{P}=3\left[\,\sqrt{\frac{1+q}{q}}\sqrt{k_{\rm disc}}\,(C_{\rm
w}+1)+\frac{2}{3}\,\frac{C_{\rm w}}{1+q}\right.-\\\
-\left.\sqrt{\frac{1+q}{q}}\sqrt{k_{\rm esc}}\,C_{\rm
w}\right]\frac{\left<\dot{M}_{\rm acc}\right>}{M_{\rm opt}}\,.\end{gathered}$
(2.15)

It should be noted that, taking into account the obtained limitation (2.13) on
the ratio of the mass loss rate in the wind to the accretion rate, the total
effect always leads to an increase in the period.

###  2.2 Flow of matter into accretion disc

Now let us turn to another source of the change in the period, namely, to the
flow of matter from the optical component of the system into the Roche lobe of
a compact object. In this case, it is also necessary to use the formula (2.2),
but now under the assumption that the parcel of mass $\Delta M_{\rm tr}$
($\Delta M_{\rm tr}>0$) is transferred from the star with point mass $M_{\rm
opt}$ and arrives into the accretion disc at the radius $R_{\rm disc}$ without
changing the total mass of the system ($\Delta M=0$):

$\frac{\Delta P_{\rm tr}}{P}=\frac{3}{2}\frac{\Delta A_{\rm tr}}{A}\,,$ (2.16)

where $\Delta P_{\rm tr}$ and $\Delta A_{\rm tr}$ are the changes in the
orbital period and semi-major axis, respectively. We also write out the
angular momentum before and after the matter transfer from the optical
component into the accretion disc:

$J=M_{\rm x}M_{\rm opt}\sqrt{\frac{GA}{M_{\rm x}+M_{\rm opt}}}\,,$ (2.17)
$\begin{gathered}J^{{}^{\prime\prime}}=M_{\rm x}\,\omega_{\rm
orb}^{{}^{\prime\prime}}\,(R_{\rm x}^{{}^{\prime\prime}})^{2}+(M_{\rm
opt}-\Delta M_{\rm tr})\,\omega_{\rm orb}^{{}^{\prime\prime}}\,(R_{\rm
opt}^{{}^{\prime\prime}})^{2}\,+\\\ +\,\Delta M_{\rm tr}(\omega_{\rm
orb}^{{}^{\prime\prime}}\,(R_{\rm x}^{{}^{\prime\prime}})^{2}+\sqrt{GM_{\rm
x}R_{\rm disc}})\,,\end{gathered}$ (2.18)

where

$\begin{gathered}R_{\rm x}^{{}^{\prime\prime}}=A^{{}^{\prime\prime}}\,(M_{\rm
opt}-\Delta M_{\rm tr})/M,\\\ R_{\rm
opt}^{{}^{\prime\prime}}=A^{{}^{\prime\prime}}\,(M_{\rm x}+\Delta M_{\rm
tr})/M,\\\ \omega_{\rm
orb}^{{}^{\prime\prime}}=\sqrt{\frac{GM}{(A^{{}^{\prime\prime}})^{3}}}\,,\\\
A^{{}^{\prime\prime}}=A+\Delta A_{\rm tr}\,,\\\ M=M_{\rm x}+M_{\rm
opt}\,.\end{gathered}$ (2.19)

In this section, by analogy with the previous one, the ’double-primed’
variables denote the state of the system after the matter transfer, and the
’unprimed’ subscript indicates the state before that.

Taking into account that $J=J^{{}^{\prime\prime}}$ (the total orbital angular
momentum does not change), as well as the expression (2.16), we obtain a
formula for changing the period of the binary system due to the flow of matter
into the disc around the compact object. Expressing everything in terms of
$\left<\dot{P}_{\rm tr}\right>$, $q$, and $\left<\dot{M}_{\rm tr}\right>$:

$\frac{\left<\dot{P}_{\rm
tr}\right>}{P}=\frac{3}{(1+q)}\left(q-\frac{1}{q}-\frac{(1+q)^{3/2}}{\sqrt{q}}\sqrt{k_{\rm
disc}}\right)\frac{\left<\dot{M}_{\rm tr}\right>}{M_{\rm opt}}.$ (2.20)

The above is reduced to the classical formula describing the flow of matter
between the point masses (see, for example, Cherepaschuk, 2013) by zeroing
$k_{\rm disc}$.

Since we consider low-mass X-ray binaries, in which a star of a lower mass is
the donor, the rate of the change in the period due to the flow of matter into
the disc is always positive ($\left<\dot{P}_{\rm tr}\right>/P>0$) and leads to
an increase in the orbital period. Note that, comparing instantaneous values,
the change in the period caused by the wind is $2-3$ orders of magnitude
higher than the change in the period due to the flow of matter.

###  2.3 Outflow from the Lagrange point $L_{2}$

As stated earlier, the outflow of matter can occur not only from the disc, but
also from the outer Lagrange $L_{2}$, which is located behind the less massive
component of the system. We use the work of Soberman et al. (1997) to estimate
the change in the period and get:

$\frac{\dot{P}_{L_{2}}}{P_{1}}=\frac{3\dot{M}_{L_{2}}}{M}\left(\frac{M^{2}}{M_{\rm
opt}\,M_{\rm x}}\sqrt{\frac{R_{L2}}{A}}\right)\,,$ (2.21)

where $\dot{M}_{L_{2}}$ ($\dot{M}_{L_{2}}<0$) — the rate of matter outflow
from the Lagrange point $L_{2}$, $R_{L_{2}}$ is the distance from $L_{2}$ to
the center of mass of the system (Emelyanov & Salyamov, 1983):

$\begin{gathered}\frac{R_{L_{2}}}{A}\approx\frac{1}{1+1/q}+\left(\frac{1/q}{3(1+1/q)}\right)^{1/3}\\!\\!\\!\\!+\frac{1}{3}\left(\frac{1/q}{3(1+1/q)}\right)^{2/3}\\!\\!\\!\\!-\\\
-\;\frac{1}{9}\left(\frac{1/q}{3(1+1/q)}\right)+\frac{50}{81}\left(\frac{1/q}{3(1+1/q)}\right)^{4/3}\,.\end{gathered}$
(2.22)

It can be seen that the outflow from the second Lagrange point leads to a
decrease in the period.

##  3 Results

Using formula (2.15), we can determine the change in the orbital period due to
the wind from the disc and accretion of matter by setting the values of the
masses of the system components, $M_{\rm x}$ and $M_{\rm opt}$, the average
rates outflow of matter and accretion $\left<\dot{M}_{\rm wind}\right>$ and
$\left<\dot{M}_{\rm acc}\right>$, disc size before the burst $R_{\rm disc}$,
as well as the effective radius of the disc $R_{\rm esc}$, from which the
outflow occurs. For the characteristic radii of the problem, it is convenient
to introduce the parameters $k_{\rm disc}\equiv R_{\rm disc}/A$ and $k_{\rm
esc}\equiv R_{\rm esc}/A$.

If the angular momentum of matter passing through the Lagrange point $L_{1}$
does not change during the formation of the disc (which is accurate enough for
typical values of $q$ in the LMXB), then the matter forms a ring with the
circulization radius $R_{\rm circ}$ rotating with the Keplerian speed. We
assume that the disc in a quiescent state before the outburst does not spread
out, and its characteristic radius $R_{\rm disc}$ remains equal to $R_{\rm
circ}$. Then, according to Frank et al. (2002) (formula 4.20):

$R_{\rm disc}=R_{\rm
circ}=\left(1+\frac{1}{q}\right)\left[0.500-0.227\lg\left(\frac{1}{q}\right)\right]^{4}\,A\,.$
(3.23)

Using (3.23), we find that, for example: $k_{\rm circ}\equiv R_{\rm
circ}/A=0.226,\;0.307$ and $0.420$ for $q=5,\;7$ and $20$, respectively.

At the beginning of an outburst, due to the fast heating of the matter in the
ring and increase in viscosity, a redistribution of the angular momentum
begins: a part of the matter with a decreasing specific angular momentum falls
along a spiral to the center, another part with a large specific angular
momentum moves away, as a result of which the disc expands to its maximum
size. In conservative accretion discs without a wind, almost all the angular
momentum from the disc is pumped from its outer radius to the orbital momentum
of the binary system (e.g., Ichikawa & Osaki, 1994).

To estimate the maximum possible effect of the influence of the wind, let us
assume that all the wind starts from the outer boundary of the disc, where the
matter has the greatest specific angular momentum. The outer boundary of the
disc is acted upon by tidal forces from the side of the neighboring component;
therefore, the accretion disc does not reach the boundaries of the Roche lobe
of the compact object. As, for example, in Suleimanov et al. (2008), we choose
the outer radius of the accretion disc equal to the tidal radius $R_{\rm
tid}$, which is about $90\%$ of the radius of the Roche lobe $R_{\rm RL}$,
defined by the Eggleton’s formula (Eggleton, 1983):

$R_{\rm esc}=R_{\rm tid}=0.9\times R_{\rm
RL}=0.9\times\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln{(1+q^{1/3})}}\ A\ ,$ (3.24)

where $q\equiv M_{\rm x}/M_{\rm opt}$ is the ratio of the masses of the
compact (for which the Roche lobe is considered) and optical components of the
close binary system. Using (3.24), we obtain: $k_{\rm tid}\equiv R_{\rm
tid}/A=0.469,\;0.520$, and $0.567$ for $q=5,\;7$, and $20$, respectively.

Note that, due to the fact that the formulas (3.23) and (3.24) are approximate
and are obtained from various considerations, for sufficiently large values of
the mass ratio $q$ (starting from $q\approx 47.8$), the circulization radius
exceeds the outer radius of the disc, which is unrealistic. Therefore, we
impose the following condition on the radius of the disc before an outburst:
$R_{\rm disc}\leq R_{\rm esc}$.

Having obtained the estimates (3.23) and (3.24) of the characteristic radii of
the problem, let us investigate how the effect of changing the period of the
binary system depends on the relative power of the wind from the disc. Let us
assume that the average accretion rate onto a compact object is
$\left<\dot{M}_{\rm acc}\right>=10^{18}$ g/s, which, by order of magnitude, is
the typical accretion rate during an LMXB outburst and is about one tenth of
the critical Eddington accretion rate onto a non-rotating black hole with mass
$M_{\rm x}=10\,M_{\odot}$.

Figure 2: Relative change in the orbital period of a close binary system
versus the ratio of the rates of mass loss due to wind and accretion for
different mass ratios if $R_{\rm esc}=R_{\rm tid}$. The average accretion rate
$\left<\dot{M}_{\rm acc}\right>$ is $10^{18}$ g/s.

Let the rate of the mass loss in the wind be proportional to the accretion
rate onto the black hole: $\left|\left<\dot{M}_{\rm wind}\right>\right|=C_{\rm
w}\times\left<\dot{M}_{\rm acc}\right>$. Fig. 2 shows the rates of the
relative change in the orbital period according to (2.15) depending on the
ratio $C_{\rm w}$ for various mass ratios of the components $q$ for $R_{\rm
esc}=R_{\rm tid}$ and $R_{\rm disc}=R_{\rm circ}$. Curves end on the right
when $C_{\rm w}=C_{\rm w}^{*}(q)$. However, for $q\gtrsim 47.8$ the value of
$C_{\rm w}^{*}$ does not exist, since the approximation formulas for the tidal
and circulation radius give a non-physical relation (see the formula (2.13)).
Therefore, the curves in the given range of values of $q$ ($q\gtrsim 47.8$)
are not limited by any maximum $C_{\rm w}$.

Note also that on the intervals $q\lesssim 0.1$ and $1.1\lesssim q\lesssim
42.7$ the curves are decreasing functions of $q$, since for these values of
$q$ the following condition is satisfied: $k_{\rm esc}^{1/2}>k_{\rm
disc}^{1/2}+2/3\,(q^{1/2}/(1+q)^{3/2})$.

Figure 3:  Normalized period variation $(\Delta P_{\rm burst}/P)\,(\Delta
M_{\rm acc}/M_{\rm opt})^{-1}$ depending on the logarithm of the component
mass ratio $q=M_{\rm x}/M_{\rm opt}$. The solid curve is the change in the
period if there is no wind from the disc ($C_{\rm w}=0$), the dashed red curve
shows the case when the mass loss in the wind is equal to the accreted mass
($C_{\rm w}=1$), but the effective radius of the wind outflow is equal to the
characteristic radius of the disc before the outburst; other curves: the wind
starts from the tidal radius of the disc $R_{\rm tid}$, $C_{\rm w}=1$ and
$C_{\rm w}=C_{\rm w}^{*}$. The area below the minimum possible period change
is shown in gray.

Figure 3 shows the value of the normalized change in the period $\frac{\Delta
P}{P}\,(\frac{\Delta M_{\rm acc}}{M_{\rm opt}})^{-1}$ depending on the mass
ratio $q$. As can be seen from the figure, the presence of the wind for the
interval of values $q$, typical for LMXB, leads to some decrease in the
predicted period gain due to the outburst, comparing with the conservative
model.

###  3.1 Applying the model to real LMXB systems

Using the code freddi111freddi can be freely downloaded from the web page
http://xray.sai.msu.ru/~malanchev/freddi. (Malanchev & Lipunova, 2016,
Lipunova & Malanchev, 2017), artificial outbursts of XTE J1118$+$480,
A0620$-$00, and GRS 1124$-$68 were simulated, taking into account disc wind
(see Fig. 4). Model parameters for systems are shown in Table. 1. Note that
the initial accretion rate for all three systems is taken equal to the
critical Eddington rate.

The freddi code calculates the viscous evolution of a disc zone that is fully
ionized. The code was developed to calculate the light curves of soft X-ray
transients with rapid growth and quasi-exponential decay and was modified for
this work to take into account the effect of wind from the disc (Avakyan et
al., 2019).

Figure 4: Simulated outbursts evolution with disc wind for parameters of XTE
J1118$+$480, A0620$-$00, and GRS 1124$-$68\. The initial accretion rate equals
to the Eddington value. The ratio of the rate of mass loss due to wind and
accretion is $C_{\rm w}=7,\,8.45,\,3.87$, respectively. The system parameters
are listed in Table 1. Table 1:  Parameters of binary systems XTE J1118$+$480,
A0620$-$00, and GRS 1124$-$68 used in simulations.

Parameters | XTE J1118$+$480 | A0620$-$00 | GRS 1124$-$68 | Ref.*
---|---|---|---|---
Black Hole Mass, $M_{\rm x}$ | $7.06M_{\odot}$ | $6.5M_{\odot}$ | $11.0M_{\odot}$ | [1], [2], [3]
Companion Star Mass, $M_{\rm opt}$ | $0.10M_{\odot}$ | $0.26M_{\odot}$ | $0.89M_{\odot}$ | [1], [2], [3]
Orbital period, $P$ |  $0.1699$ d |  $0.3230$ d |  $0.4326$ d | [4], [5], [6]
Inclination, $i$ | $74.0^{\circ}$ | $51.0^{\circ}$ | $43.2^{\circ}$ | [2], [7], [3]
Viscosity parameter, $\alpha$ | $0.1$ | $0.1$ | $0.1$ | [8]

* [1] Cherepashchuk et al. (2019b); [2] Cherepashchuk et al. (2019a); [3] Wu et al. (2016), [4] GH12; [5] GH14; [6] GH17; [7] Cantrell et al. (2010); [8] Shakura & Sunyaev (1973).
Note. The peak accretion rate is equal to the Eddington one: $\dot{M}_{\rm
acc,0}=\dot{M}_{\rm Edd}=1.4\times 10^{18}\ (M_{\rm x}/M_{\odot})$ g/s.

The accretion rates obtained by modeling were used to find an estimate of the
change in the orbital period (Figs. 5, 6 and 7). The ratio of the rate of mass
loss in the wind to the accretion rate onto the black hole is chosen as
$C_{\rm w}=7;\,8.45;\,3.87$ for XTE J1118$+$480, A0620$-$00 and GRS 1124$-$68,
respectively. The maximum possible $C_{\rm w}=C_{\rm w}^{*}(q)$ is taken for
A0620$-$00 and GRS 1124$-$68, appplying $R_{\rm esc}=R_{\rm tid}$ and $R_{\rm
disc}=R_{\rm circ}$. Such values lead to the lowest estimate of $\Delta P$.
For parameters of XTE J1118$+$480, the definition of $C_{\rm w}^{*}$ is not
applicable due to the formal equality of the tidal radius $R_{\rm tid}$ and
the circulization radius $R_{\rm circ}$ (i.e., $k_{\rm esc}=k_{\rm disc}$).
This happens because the value of the mass ratio of the components in this
system $q$ exceeds the boundary value of $47.8$. Therefore, $C_{\rm w}$ was
set equal to 7 (Luketic et al., 2010). The lowest $\Delta P$ for this system
is realized when $C_{\rm w}=0$ (see Fig. 3).

Figure 5: Change in the period for the XTE J1118$+$480 (within one year)
according to the model (blue curves: mass transfer and outburst), observations
(black), and constant outflow from the Lagrange point $L_{2}$ (green and red).
Figure 6: Change in the period for the A0620$-$00 (within one year) according
to the model (blue curves: mass transfer and outburst), observations (black),
and constant outflow from the Lagrange point $L_{2}$ (green and red). Figure
7: Change in the period for the GRS 1124$-$68 (within one year) according to
the model (blue curves: mass transfer and outburst), observations (black), and
constant outflow from the Lagrange point $L_{2}$ (green and red).

With these parameters, the average values of the orbital period change during
the outburst are about $4.3\times 10^{-10}$ s/s, $9.6\times 10^{-11}$ s/s, and
$8.2\times 10^{-11}$ s/s for XTE J1118$+$480, A0620$-$00, and GRS 1124$-$68,
respectively. Such a strong change of the orbital period occurs only during
the outburst itself (about $40-90$ days, we chose $45$ for XTE J1118$+$480 and
A0620$-$00, and $85$ days for GRS 1124$-$68). In a steady state (lasting from
several to tens of years), the accretion and wind rates in the disc are much
lower, and the mechanisms of wind formation can be completely switched off.

Table 2 shows the calculated values of the period variations due to the wind
accretion for the XTE J1118$+$480, A0620$-$00, and GRS 1124$-$68 under the
assumption of one outburst per $\Delta T_{\rm q}=6,\ 30$, and $20$ years,
respectively. In other words, the systems, according to the model, are in a
state with high accretion rate for several weeks (Fig. 4), when the period is
changing significantly, after which a quiescent state begins, lasting $\Delta
T_{\rm q}$. Specific values $\Delta T_{\rm q}$ are not chosen by chance. XTE
J1118$+$480 bursts quite often, in 2000 and 2005 (Šimon, 2020), which cannot
be said about A0620$-$00 and GRS 1124$-$68, in which the outbursts have not
been observed since 1975 and 1991 (Connors et al., 2017, Wu et al., 2016),
respectively. The observations of XTE J1118$+$480, used in the works of GH12
and GH14 to determine the decrease in the orbital period, were carried out
from 2000 to 2012. In the case of A0620$-$00, the last measurement of the
period was made in 2006 (GH14), and the last outburst occurred in 1975. For
GRS 1124$-$68, the last measurement of the period was in 2012 (GH17), more
than twenty years after the outburst in 1991.

Table 2:  Orbital changes $\Delta P_{\rm burst}/\Delta T_{\rm q}$ [s/s] due to a ‘windy’ outburst for XTE J1118$+$480, A0620$-$00, and GRS 1124$-$68 as well as the observed and model values of $\dot{P}$ [s/s] from GH and CP19. Parameters | XTE J1118$+$480 | A0620$-$00 | GRS 1124$-$68
---|---|---|---
Our lower estimate | $6.6\times 10^{-11}$ | $3.9.\times 10^{-13}$ | $9.5\times 10^{-13}$
Observations (GH) | $-6.0\times 10^{-11}$ | $-1.9\times 10^{-11}$ | $-6.6\times 10^{-10}$
Model 2 (GH) | $-5.4\times 10^{-13}$ | $-7.6\times 10^{-13}$ | $-8.9\times 10^{-13}$
Model 1 (GH) | $-2.7\times 10^{-11}$ | $-8.6\times 10^{-12}$ | $-3.5\times 10^{-12}$
Gravitational waves (CP19) | $-3.0\times 10^{-13}$ | $-2.0\times 10^{-13}$ | $-4.0\times 10^{-13}$
Magnetic braking (CP19) | $-7.8\times 10^{-12}$ | $-3.8\times 10^{-12}$ | $-2.2\times 10^{-12}$

In addition to our results, Table 2 shows long-term values of $\dot{P}$ from
the series of works GH12, GH14, and GH17 (both theoretical and determined from
observations, hereinafter referred to as ’GH’) and estimates of CP19. In the
works of the GH, the model of the period decrease of a binary by Johannsen et
al. (2009) is used, which takes into account the magnetic braking and loss of
matter due to the evaporation of a black hole. The GH uses two sets of
parameters for this model, namely, ‘realistic’ and ‘extreme’ (the maximum
possible effect of mass loss and magnetic braking). In Table 2, the
designations Model 1 and Model 2 are chosen for the ‘realistic’ and ‘extreme’
set of parameters, respectively. All values are given in order to
qualitatively compare the power of various mechanisms that affect the
evolution of the orbital periods of binary systems.

Period changes due to outbursts are hardly noticeable against the background
of the observed secular trend in XTE J1118$+$480, A0620$-$00 and GRS 1124$-$68
on scales of 6, 30, and 20 years, respectively. Note that the values in the
first line of Table 2 do not take into account other effects that cause
secular changes in the orbital period described above, namely, the outflow
from the Lagrange point $L_{2}$ and the transfer of the matter from the
optical component to the disc.

Figures 5, 6, and 7 show the period variations in a year when an outburst
occurs. Lines with a ‘step’ show the total effect, namely, the sum of the
mechanisms of the matter transfer (a subtle secular increase in the period for
$\left<\dot{M}_{\rm tr}\right>=10^{16}$ g/s) and the variation during the
outburst (the step itself). The effect of the outflow from the point $L_{2}$
is depicted for the two values of the mass loss rate, differing by a factor of
$10$. The outflow from the point $L_{2}$ leads to a secular decrease in the
period, but the observed rate requires an abnormally high rate of the mass
loss. It can be seen that for each of the three systems, the outflow from
$L_{2}$ could explain the decrease in the period of the order of the observed
one only with very large, unrealistic loss rates through $L_{2}$, especially
in the case of GRS 1124$-$68.

Observations of the orbital period of an X-ray binary before and after an
X-ray nova outburst would be a good evidence in favor of a strong period
change during an outburst. According to our model, the orbital periods of XTE
J1118$+$480, A0620$-$00, and GRS 1124$-$68 can increase by $\gtrsim 1.6$,
$\gtrsim 0.4$, and $\gtrsim 0.6$ ms, respectively, however, such observations
of the sources have not yet been carried out.

##  4 Discussion

According to the observations of GH14 and GH17, the three considered LMXB
systems with black holes demonstrate the strong period decrease in the
quiescent state. The decrease in the period due to the gravitational radiation
(CP19) is $2-3$ orders of magnitude less than the observed rate, and estimates
for the magnetic braking are less by an order of magnitude, even with very
favorable parameters (see Table 2). Of the mechanisms discussed above, only
the outflow of matter from point $L_{2}$ with an excessively high rate could
explain the observed rates (see Fig. 5, 6, and 7).

In addition to the mechanisms discussed so far, the following scenario is
possible. The matter ejected by the wind from the accretion disc, or escaping
through the $L_{2}$ point, can form a toroidal disc around the binary system
(a so-called circumbinary disc). Due to the viscosity processes, the
circumbinary disc can drain the moment of tidal forces acting on it, and
thereby, reduce the angular momentum of the binary system (therefore
decreasing its orbital period). The characteristic viscous time of evolution
of a circumbinary disc significantly exceeds the viscous time of the hot disc
around a compact object, which allows it to remove the angular momentum on a
longer time scale. Muno & Mauerhan (2006) and Wang & Wang (2014) provide
observational evidence for the existence of such a disc around A0620$-$00 and
XTE J1118$+$480\. In the work of CP19, based on the results of Artymowicz &
Lubow (1994), the influence of such a ring on the period of a close binary
system has been studied. According to CP19, the observed decrease in the
period of A0620$-$00 and XTE J1118$+$480 can be explained if the mass of the
circumbinary disc is approximately equal to $10^{-9}\,M_{\odot}$. It follows
from our simulations that for A0620$-$00 and XTE J1118$+$480, during an
outburst (even for the case $C_{\rm w}=2$), the mass taken with the wind from
the disc is, respectively, $3.4\times 10^{-8}\,M_{\odot}$ and $2.4\times
10^{-8}\,M_{\odot}$. Most likely, not all the matter of the wind settles into
a ring around the binary, but even if so, such an estimate may be sufficient
to explain the observed decrease in the period for the A0620$-$00 and XTE
J1118$+$480\. However, for GRS 1124$-$68, the ring around the binary system
must be two orders of magnitude heavier, namely, $10^{-7}\,M_{\odot}$. The
required mass can be accumulated after one outburst, if the wind is very
strong (for $C_{\rm w}=10$ the mass of matter ejected in the wind is
$1.1\times 10^{-7}\,M_{\odot}$). The fact of such a strong change in the GRS
1124$-$68 period makes it even more interesting for a further study.

##  5 Summary

An outburst in a low-mass X-ray binary leads to a significant change of its
orbital period. We have obtained a general analytical formula for assessing
this effect, taking into account the wind from the accretion disc, and have
given the quantitative estimates of the magnitude of the period change for
three LMXBs.

Observations of various LMXB systems and measurements of the orbital period
immediately before and after outbursts can reveal the predicted changes in the
orbital period and provide additional information concerning the rate of
matter loss due to wind.

We have also considered the changes in the orbital period due to mechanisms
operating in the quiescent state of LMXBs: the transfer of matter from the
donor to the disc around the black hole and the outflow of matter from the
point $L_{2}$. The latter leads to a secular decrease in the orbital period.
However, it also cannot explain the observed rate of the period decrease in
the sources considered. Apparently, it is necessary to involve either
extremely strong magnetic braking, or the drain of the orbital angular
momentum by a ring of matter surrounding a binary system.

## Acknowledgements

The authors are grateful to K.A. Postnov for valuable remarks, the seminar of
the Relativistic Astrophysics Department for a productive discussion, and also
to I. I. Antokhin for comments. ALA is grateful to Foundation for the
Advancement of Theoretical Physics and Mathematics ‘BASIS’ for supporting his
work (grant number 20-2-1-106-1). This research has been supported by the
Interdisciplinary Scientific and Educational School of Moscow University
“Fundamental and Applied Space Research”. The development of the code freddi
was supported by the RFBR grant no. 18-502-12025.

## References

  * Artymowicz & Lubow (1994) Artymowicz P., Lubow S. H., 1994, ApJ, 421, 651
  * Avakyan et al. (2019) Avakyan A. L., Malanchev K. L., Lipunova G. V., 2019, in The Multi-Messenger Astronomy: Gamma-Ray Bursts, Search for Electromagnetic Counterparts to Neutrino Events and Gravitational Waves. pp 25–31, doi:10.26119/SAO.2019.1.35452
  * Bath (1975) Bath G. T., 1975, MNRAS, 171, 311
  * Bath & Pringle (1981) Bath G. T., Pringle J. E., 1981, MNRAS, 194, 967
  * Bath et al. (1974) Bath G. T., Evans W. D., Papaloizou J., Pringle J. E., 1974, MNRAS, 169, 447
  * Cantrell et al. (2010) Cantrell A. G., et al., 2010, ApJ, 710, 1127
  * Casares et al. (2019) Casares J., Muñoz-Darias T., Mata Sánchez D., Charles P. A., Torres M. A. P., Armas Padilla M., Fender R. P., García-Rojas J., 2019, MNRAS, 488, 1356
  * Charles et al. (2019) Charles P., Matthews J. H., Buckley D. A. H., Gandhi P., Kotze E., Paice J., 2019, MNRAS, 489, L47
  * Chen & Podsiadlowski (2019) Chen W.-C., Podsiadlowski P., 2019, ApJ, 876, L11
  * Chen et al. (1997) Chen W., Shrader C. R., Livio M., 1997, ApJ, 491, 312
  * Cherepaschuk (2013) Cherepaschuk A. M., 2013, Close Binary Stars, II. MOSCWA FIZMATLIT
  * Cherepashchuk et al. (2019a) Cherepashchuk A. M., Katysheva N. A., Khruzina T. S., Shugarov S. Y., Tatarnikov A. M., Burlak M. A., Shatsky N. I., 2019a, MNRAS, 483, 1067
  * Cherepashchuk et al. (2019b) Cherepashchuk A. M., Katysheva N. A., Khruzina T. S., Shugarov S. Y., Tatarnikov A. M., Bogomazov A. I., 2019b, MNRAS, 490, 3287
  * Connors et al. (2017) Connors R. M. T., et al., 2017, MNRAS, 466, 4121
  * Díaz Trigo & Boirin (2016) Díaz Trigo M., Boirin L., 2016, Astronomische Nachrichten, 337, 368
  * Eggleton (1983) Eggleton P. P., 1983, ApJ, 268, 368
  * Emelyanov & Salyamov (1983) Emelyanov N. V., Salyamov V. N., 1983, Soviet Ast., 27, 442
  * Frank et al. (2002) Frank J., King A., Raine D. J., 2002, Accretion Power in Astrophysics: Third Edition.
  * González Hernández et al. (2012) González Hernández J. I., Rebolo R., Casares J., 2012, in Richards M. T., Hubeny I., eds, Vol. 282, From Interacting Binaries to Exoplanets: Essential Modeling Tools. pp 476–477, doi:10.1017/S174392131102816X
  * González Hernández et al. (2014) González Hernández J. I., Rebolo R., Casares J., 2014, MNRAS, 438, L21
  * González Hernández et al. (2017) González Hernández J. I., Suárez-Andrés L., Rebolo R., Casares J., 2017, MNRAS, 465, L15
  * Hameury (2020) Hameury J. M., 2020, Advances in Space Research, 66, 1004
  * Higginbottom & Proga (2015) Higginbottom N., Proga D., 2015, ApJ, 807, 107
  * Higginbottom et al. (2017) Higginbottom N., Proga D., Knigge C., Long K. S., 2017, ApJ, 836, 42
  * Higginbottom et al. (2019) Higginbottom N., Knigge C., Long K. S., Matthews J. H., Parkinson E. J., 2019, MNRAS, 484, 4635
  * Ichikawa & Osaki (1994) Ichikawa S., Osaki Y., 1994, PASJ, 46, 621
  * Jeans (1928) Jeans J. H., 1928, Cambridge: Cambridge University Press, pp 280–290
  * Johannsen et al. (2009) Johannsen T., Psaltis D., McClintock J. E., 2009, ApJ, 691, 997
  * Kosec et al. (2020) Kosec P., Fabian A. C., Pinto C., Walton D. J., Dyda S., Reynolds C. S., 2020, MNRAS, 491, 3730
  * Kruszewski (1964) Kruszewski A., 1964, Acta Astron., 14, 241
  * Landau & Lifshitz (1962) Landau L. D., Lifshitz E. M., 1962, Physics Today, 15, 48
  * Lasota (2016) Lasota J.-P., 2016, Black Hole Accretion Discs. Springer, p. 1, doi:10.1007/978-3-662-52859-4˙1
  * Lipunova & Malanchev (2017) Lipunova G. V., Malanchev K. L., 2017, MNRAS, 468, 4735
  * Luketic et al. (2010) Luketic S., Proga D., Kallman T. R., Raymond J. C., Miller J. M., 2010, ApJ, 719, 515
  * Malanchev & Lipunova (2016) Malanchev K. L., Lipunova G. V., 2016, Freddi: Fast Rise Exponential Decay accretion Disk model Implementation, Astrophysics Source Code Library (ascl:1610.014)
  * Muñoz-Darias et al. (2019) Muñoz-Darias T., et al., 2019, ApJ, 879, L4
  * Muno & Mauerhan (2006) Muno M. P., Mauerhan J., 2006, ApJ, 648, L135
  * Ponti et al. (2012) Ponti G., Fender R. P., Begelman M. C., Dunn R. J. H., Neilsen J., Coriat M., 2012, MNRAS, 422, L11
  * Rappaport et al. (1982) Rappaport S., Joss P. C., Webbink R. F., 1982, ApJ, 254, 616
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
  * Soberman et al. (1997) Soberman G. E., Phinney E. S., van den Heuvel E. P. J., 1997, A&A, 327, 620
  * Suleimanov et al. (2008) Suleimanov V. F., Lipunova G. V., Shakura N. I., 2008, A&A, 491, 267
  * Tutukov (1983) Tutukov A. V., 1983, Pisma v Astronomicheskii Zhurnal, 9, 160
  * Tutukov & Yungelson (1971) Tutukov A. V., Yungelson L. R., 1971, Nauchnye Informatsii, 20, 86
  * Ueda et al. (2009) Ueda Y., Yamaoka K., Remillard R., 2009, ApJ, 695, 888
  * Verbunt & Zwaan (1981) Verbunt F., Zwaan C., 1981, A&A, 100, L7
  * Wang & Wang (2014) Wang X., Wang Z., 2014, ApJ, 788, 184
  * Wu et al. (2016) Wu J., Orosz J. A., McClintock J. E., Hasan I., Bailyn C. D., Gou L., Chen Z., 2016, ApJ, 825, 46
  * Šimon (2020) Šimon V., 2020, PASA, 37, e003