# Stochastic control of spiking activity bump expansion: monotonic and
resonant phenomena

Anna Zakharova  Institut für Theoretische Physik, Technische Universität
Berlin, 10623 Berlin, Germany  Bernstein Center for Computational
Neuroscience, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin,
Germany  Vladimir V. Semenov<EMAIL_ADDRESS>Institut für
Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
Institute of Physics, Saratov State University, 410012 Saratov, Russia

###### Abstract

We consider spatially localized spiking activity patterns, so-called bumps, in
ensembles of bistable spiking oscillators. The bistability consists in the
coexistence of self-sustained spiking dynamics and quiescent steady-state
regime. We show numerically that the processes of growth or contraction of
such patterns can be controlled by varying the intensity of multiplicative
noise. In particular, the effect of the noise is monotonic in an ensemble of
the coupled Hindmarsh-Rose oscillators. On the other hand, in another model
proposed by V. Semenov et al. in 2016 (see Ref. Semenov  _et al._ (2016)), a
resonant noise effect is observed. In that model, stabilization of the
activity bump expansion is achieved at an appropriate noise level, and the
noise effect reverses with a further increase in noise intensity. Moreover, we
show the constructive role of nonlocal coupling which allows to save domains
and fronts being totally destroyed due to the action of noise in the case of
local coupling.

activity bumps, bistability, front propagation, coarsening, stochastic control

###### pacs:

05.10.-a, 05.45.-a, 05.40.Ca

††preprint: AIP/123-QED

##  I Introduction

Formation of self-sustained states localized in space, so-called localised
activity bumps, is a frequent phenomenon in biological neural networks Coombes
_et al._ (2014). In particular, such states have been reported by
neurophysiologists and associated with such brain functions as working memory
Compte  _et al._ (2000); Wimmer  _et al._ (2014) and head-direction control
Taube (1998); Sharp, Blair, and Cho (2001). A broad spectrum of mathematical
models is used for the investigation of such structures in different forms:
stationary soliton solution Blomquist, Wyller, and Einevoll (2005); Yousaf
_et al._ (2013), localised oscillating structures Rubin and Bose (2004); Rubin
(2005); Rubin and Bose (2006); Laing (2015); Avitabile and Wedgwood (2017);
Schmidt and Avitabile (2020), propagating and expanding patterns Avitabile and
Schmidt (2015). In the present work, the existence and expansion of spiking
activity bumps in ensembles of bistable oscillators are explored. Here, the
property of bistability in the studied models results from the coexistence of
a stable limit cycle and a stable steady state in the phase space of an
individual (uncoupled) oscillator.

It is well known that bistable spatially-extended systems can exhibit dynamics
in which two domains with propagating fronts between them are formed in space.
Such fronts occur frequently in chemistry, for instance, in the Schlögl model
Schlögl (1972); Schlögl, Escher, and Berry (1983); Mikhailov and Ertl (2014)
developed to explain an autocatalytic reaction mechanism, as well as in
various other fields, such as electronics Schöll (2001) or flame propagation
theory Zel’dovich and Frank-Kamenetskii (1938), to name just a few. Similar
processes associated with domain expansion and called ’coarsening’ are studied
in the context of physics of liquid crystals Yurke  _et al._ (1992), magnetism
Bray (1994); Cugliandolo (2010); Caccioli, Franz, and Marsili (2008); Denholm
and Redner (2019), physics and chemistry of materials Goh and Jacobs (2002);
Zhang, Yu, and Wang (2019); Zhang, Liu, and Crozier (2019); Geslin  _et al._
(2019), laser physics Giacomelli  _et al._ (2012); Marino, Giacomelli, and
Barland (2014); Javaloyes, Ackemann, and Hurtado (2015), electronics Semenov
and Maistrenko (2018), and animal population statistics Dobramysl  _et al._
(2018). It should be noted that besides bistable spatially-extended systems
Bray (1994), such effects have been observed for time-delay models Giacomelli
_et al._ (2012); Marino, Giacomelli, and Barland (2014); Semenov and
Maistrenko (2018); Ruschel and Yanchuk (2019).

Since the phenomena referred to as wavefront propagation and coarsening are
observed in a wide spectrum of dynamical systems, the development of
principles to control these effects represents a relevant interdisciplinary
problem. One of the known approaches involves the impact of noise for this
purpose. Noise is actively used for the control of various phenomena such as,
for example, synchronization in multilayer networks where the so-called
multiplexing noise included into the interaction of the network layers
regulates the synchronization of spatio-temporal patterns Vadivasova, Slepnev,
and Zakharova (2020). Moreover, constructive role of noise and the related
resonance phenomena continue to attract the attention of researches as
confirmed by recent works on, for example, coherence resonance Masoliver  _et
al._ (2017); Bogatenko and Semenov (2018); Semenova and Zakharova (2018);
Pisarchik  _et al._ (2019); Yamakou and Jost (2019); Masoliver, Masoller, and
Zakharova (2021) or stochastic resonance Semenov and Zakharova (2022); Yamakou
and Tran (2021).

In the context of front propagation in bistable media, an effective control is
achieved by applying multiplicative noise Engel (1985) and relies on the fact
that multiplicative noise has a systematic impact on the symmetry properties
of the reaction-term Garcia-Ojalvo and Sancho (1999); Méndez  _et al._ (2011).
This multiplicative-noise-based scheme is also used in the current study.

We show that the effects of noise on propagating phenomena take more
complicated forms in bistable systems where oscillatory and steady-state
dynamics coexists compared to bistable systems with two coexisting stable
steady states, whereas the ability of multiplicative noise to control the
propagation speed and direction of the front remains. In particular, we
demonstrate numerically that the noise effect can be either monotonic or
resonant, depending on the intrinsic properties of a partial oscillator in the
ensemble. For this purpose, we consider two examples. The first example
exhibits a monotonic effect of noise, where the increase in noise intensity
first slows and stops the front propagation and then reverses its direction.
In contrast, the effect observed in the second model has a resonant character,
where the front propagation is slowed down and stabilized due to the action of
noise in a finite range of noise intensity.

##  II Monotonic impact of noise

First, we consider a ring of the nonlocally coupled Hindmarsh-Rose oscillators
(schematically illustrated in Fig. 1(a)) with multiplicative noise. A two-
dimensional modification of the Hindmarsh-Rose oscillator described in Ref.
Tsuji  _et al._ (2007) was chosen as a partial oscillator:

$\begin{array}[]{rl}\varepsilon\dfrac{dx_{i}}{dt}=&\left(a+\sqrt{2D}n_{i}(t)\right)x_{i}-\dfrac{x^{3}_{i}}{3}-y_{i}\\\
&+\dfrac{\sigma}{2R}\sum\limits^{i+R}_{j=i-R}(x_{j}-x_{i}),\\\
\dfrac{dy_{i}}{dt}=&x^{2}_{i}+bx_{i}-cy_{i},\end{array}$ (1)

where $\varepsilon$, $a$, $b$, $c$ are the partial oscillator’s parameters,
$R$ is a coupling radius, $\sigma$ is the coupling strength, $i=1,...,N$.
Further, $n_{i}(t)\in\mathbb{R}$ is Gaussian white noise, i.e. $\langle
n_{i}(t)\rangle=0$ and $\langle
n_{i}(t)n_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$ for all
$i,j$, and $D$ is the noise intensity. The ensemble is studied for periodic
boundary conditions (ring), i.e., the indices are considered modulo $N$.
Numerical modelling of ensemble (1) and further model (2) is carried out by
integration of studied differential equations using the Heun method Mannella
(2002) with the time step $\Delta t=0.0001$.

In this work we fix the parameters $\varepsilon=0.05$, $a=1$, $b=1.9$, and
$c=0.55$, for which the partial oscillator represents a bistable dynamical
system. The corresponding structure of the partial oscillator phase space is
depicted in Fig. 1(b). Two attractors coexist in the phase space: a stable
limit cycle (red closed curve) and a stable steady state (blue circle), whose
basins of attraction (translucent blue and red regions) are separated by the
separatrix (black arrowed curves) of the saddle (light grey circle). Two
nullclines ${dx}/{dt}=0$ (blue solid line) and ${dy}/{dt}=0$ (magenta dashed
line) intersect such that there exists the third equilibrium point (dark grey
circle) which is unstable.

Figure 1: (a) Schematic representation of a ring of nonlocally coupled
oscillators; (b)-(e) System Eqs. (1): (b) Partial oscillator’s phase space for
$\varepsilon=0.05$, $a=1$, $b=1.9$, $c=0.55$. Two attractors coexist: a stable
limit cycle (red closed curve) and a stable equilibrium point (blue circle).
The translucent red and blue regions display the basin of attraction for the
limit cycle and the equilibrium point, respectively. The light grey circle
represents a saddle fixed point, black arrowed curves indicate the saddle
separatrix. The dark grey circle marks an unstable steady state. Nullclines
are shown by the blue solid ($dx/dt=0$) and magenta dashed ($dy/dt=0$) curves;
(c)-(e) Space-time plots illustrating propagating phenomena for varying noise
intensity for the fixed coupling strength $\sigma=0.3$ and coupling radius
$R=10$. The partial oscillator parameters are the same as for panel (b). The
upper insets show the ensemble state at the last moment $t=10^{3}$.

The ensemble Eqs. (1) of $N=500$ oscillators is studied for the bistable
regime and the fixed coupling radius $R=10$. If the initial conditions are
chosen such that both basins of attraction of the stable limit cycle and the
equilibrium point are involved, then two kinds of domains corresponding to the
self-oscillatory spiking activity and the quiescent steady state regime are
formed. After that, fronts separating the domains begin to propagate such that
the domains of spiking activity invade the entire space [Fig. 1 (c)]. The
instantaneous state of system Eqs. (1) is depicted by black points in [Fig. 1
(b)]. Thus, the collective spiking activity of oscillators is associated with
the slow-fast motions tracing trajectories close to the stable limit cycle
existing in the phase space of the single oscillator. As can be seen in Fig. 1
(c), the collective behaviour in spiking activity domains has asynchronous
character which is a manifestation of spatial chaos.

In the presence of multiplicative noise the front propagation speed depends on
the noise intensity. As illustrated in Fig. 1 (c)-(e), it slows down with the
noise intensity increasing, so that the activity bump expansion stops for a
certain noise level, see Fig. 1 (d). Further increasing of the noise strength
inverts the front propagation and the quiescent steady state regime invades
the whole space Fig. 1 (e). To visualise the impact of noise, we introduce the
front propagation speed as $v=(w(t)-w(t+\Delta t))/2\Delta t$. Here, $w(t)$
and $w(t+\Delta t)$ are the activity bump widths at the moments $t$ and
$t+\Delta t$, respectively (illustrated in Fig. 2 (a)). Figure 2 (b) shows the
dependence of front velocity on the noise intensity $D$ for different values
of the coupling range $R$. Let us discuss the curve $v(D)$ for $R=10$. First,
the dependence $v(D)$ is almost linear (in the range $D\in[0,0.002]$). After
that, the dependence becomes almost flat, such that the propagation speed
tends to zero in the range $D\in[0.0035,0.0046]$ (grey region in Fig. 2 (b)).
The negative values of $v(D)$ mean that the front propagation is inverted.

Figure 2: Dynamics in system Eqs. (1): (a) Illustration for the calculation of
the front propagation speed $v$ on an example of a single domain; (b)
Dependence of the front propagation speed on the noise intensity $v(D)$
obtained for different values of the coupling range $R$; (c)-(f) Spatio-
temporal dynamics for the fixed value of noise intensity $D=0.006$, the same
initial conditions and different values of the coupling radius $R$. The space-
time plots in panels (a) and (c)-(f) are obtained in numerical simulations
starting from the same initial conditions. The upper insets show the ensemble
state at the last moments $t=10^{3}$ or $t=10^{4}$. Other parameters:
$\varepsilon=0.05$, $a=1$, $b=1.9$, $c=0.55$, $\sigma=0.3$.

Another interesting observation is that non-local coupling can play a
constructive role in stabilizing fronts. This is manifested as sustaining the
propagating fronts for a higher noise level, whereas the fronts collapse in
the presence of local coupling. This is shown in Fig. 2(c)-(f) where the
space-time plots obtained for $D=0.006$ demonstrate the front propagation
starting from the same initial conditions. In the case of local coupling,
$R=1$, stochastic force destroys the initial dynamics such that new domains
and fronts eventually appear and disappear [Fig. 2(c)]. Increasing the
coupling radius allows to save the initial fronts and domains, which didn’t
persist before [Fig. 2 (d)] (for this reason, the curves in Fig. 2 (b) are
built for different noise intensity ranges, the blue curve corresponds to the
longest range). Additionally, in the presence of non-local coupling we
observed the collapse of domains characterized by the width comparable with
the coupling radius. Thus, for larger coupling radius the exhibited front
propagation finishes earlier [Fig. 2 (e),(f)].

##  III Resonant phenomenon

The model Eqs. (1) has the following specific properties. First, the the
regime of bistability such as depicted in Fig. 1(b) can be achieved in a short
parameter range. Second, nullclines in Fig. 1(b) are located very close to
each other in the left part of the phase space. For these reasons, the
localized activity bump control described above might be rather sensitive to
the parameter changes.

Figure 3: Dynamics in system Eqs. (2): (a) Partial oscillator’s phase space
for $\varepsilon=0.01$, $a_{1}=1$, $a_{3}=9$, $a_{5}=22$, $b=2.5$, $c=20$,
$d=150$. Two attractors coexist: a stable limit cycle (red closed curve) and a
stable equilibrium point (blue circle). The translucent red and blue regions
display the basin of attraction for the limit cycle and the equilibrium point,
respectively. The light grey circle represents a saddle fixed point, black
arrowed curves indicate the saddle separatrix. The dark grey circle marks an
unstable steady state. Nullclines are shown by the blue solid ($dx/dt=0$) and
magenta dashed ($dy/dt=0$) curves; (b)-(e) Space-time plots illustrating
propagating phenomena for varying noise intensity for the fixed coupling
strength $\sigma=0.7$ and coupling radius $R=10$. Other parameters are the
same as in panel (a). The upper insets show the ensemble state at the last
moment $t=10^{3}$ on an example of the variables $x_{i}$; (f) Dependence of
the front propagation speed on the noise intensity $v(D)$ for different values
of coupling radius $R$. Partial oscillator parameters are the same as for
panel (a), the coupling strength is $\sigma=0.7$.

To demonstrate the robustness of the activity bump control, as well as its
different manifestation, we consider the second model: an ensemble of modified
bistable spiking oscillators initially discussed in the symmetric form in
Refs. Semenov  _et al._ (2016); Semenov (2017), where the resonant stochastic
phenomena in the chosen oscillator were demonstrated. In this work, we
introduce an asymmetry into the system to achieve the coexistence of a stable
limit cycle and a stable steady state. The ensemble under study admits the
following form:

$\begin{array}[]{rl}\varepsilon\dfrac{dx_{i}}{dt}=&-\left(1+\sqrt{2D}n_{i}(t)\right)y_{i}-a_{1}x_{i}\\\
&+a_{3}x_{i}^{3}-a_{5}x_{i}^{5},\\\
\dfrac{dy_{i}}{dt}=&x_{i}+by_{i}-cy_{i}^{2}-dy_{i}^{3}+\dfrac{\sigma}{2R}\sum\limits^{i+R}_{j=i-R}(y_{j}-y_{i}).\end{array}$
(2)

Here $\varepsilon$, $a_{1,3,5}$, $b$, $c$, and $d$ are the partial
oscillator’s parameters, $R$ is a coupling radius, $\sigma$ is the coupling
strength, $i=1,...,N$, $n_{i}(t)\in\mathbb{R}$ is Gaussian white noise, $D$ is
the noise intensity. The partial oscillator parameters are fixed as follows
$\varepsilon=0.01$, $a_{1}=1$, $a_{3}=9$, $a_{5}=22$, $b=2.5$, $c=20$,
$d=150$, for which the partial oscillator exhibits the coexistence of a stable
limit cycle (red closed curve in Fig. 3 (a)) and a stable fixed point (blue
circle in Fig. 3(a)). There is no qualitative difference between the phase
space structure of the two considered models in Fig. 1(b) and Fig. 3(a). The
main quantitative difference is the proximity of the attractors to the
nullcline and the nullclines to each other.

Spatio-temporal dynamics of ensemble Eqs. (2) for initial conditions involving
both basins of attraction consists in propagating fronts. This results in the
extension of domains corresponding to self-oscillatory motions along the limit
cycle (see the black points in Fig. 3 (a) illustrating the instantaneous state
of the ensemble) such that the spiking activity domains invade the entire
available space [Fig. 3 (b)].

Depending on the way the noise is introduced, stochastic control of the
wavefront propagation manifests itself in different ways. We consider here the
specific stochastic term $y_{i}(t)\sqrt{2D}n_{i}(t)$ (represented by
multiplicative noise source, see Eqs. (2)), since in this case particularly
interesting features of stochastic control are revealed. First, the increase
in noise intensity changes the front propagation speed, slowing down the
spread of the spiking activity bump [Fig. 3(c)]. Interestingly, in contrast to
the phenomenon demonstrated in the Sec. II, further increasing the noise level
in ensemble Eqs. (2) speeds up the front propagation [Fig. 3 (d)], which
allows to obtain the localised activity bump expansion being faster than that
in the deterministic system (compare Figs. 3(b) and 3(e)). Thus, we observe a
resonant effect: there is a certain noise intensity range where the front
propagation velocity achieves minimum. This is illustrated in Fig. 3(f), where
the dependence of the wavefront propagation speed on the noise intensity
$v(D)$ is shown.

In contrast to system Eqs. (1), the ensemble Eqs. (2) does not exhibit noise-
induced front destruction for a wide range of the noise intensity, and an
increase of the coupling radius leads only to quantitative changes. Here, the
growth of the coupling radius enhances the front propagation, while the
resonant character of the observed effect persists [Fig. 3(f)].

##  IV Conclusions

We have demonstrated that propagation phenomena in ensembles of non-locally
coupled bistable spiking oscillators can be controlled by multiplicative
noise. The studied noise-induced spatio-temporal dynamics is similar to the
noise-controlled wavefront propagation in classical ensembles with local
coupling and bistable media exhibiting the coexistence of two stable steady
states. We show that the noise-sustained control of fronts separating
localized spiking activity bumps can have a monotonic character, i.e., the
wavefront propagation velocity depends monotonically on the noise intensity.
Interestingly, it can have a resonant character as well, i.e., there exists a
certain interval of noise intensity values where the front propagation
velocity reaches its minimum.

The results obtained here show the constructive role of nonlocal coupling in
preserving the propagating wavefront patterns from being destroyed by noise.
This fact is an intriguing topic and will be the subject of future studies.

Certain processes in biological neuronal networks are well described by
bistable models (see, for instance, bistable models for epileptic seizure
description da Silva  _et al._ (2003); Suffczynski, Kalitzin, and da Silva
(2004)). For this reason, the proposed approach for stochastic control of
activity bump expansion can be potentially useful in the context of
neurophysiology and medicine.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgements

The authors sincerely acknowledge Serhiy Yanchuk for fruitful discussions and
his efforts in providing valuable and helpful advice on all stages of the
manucsript preparation. The authors acknowledge support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer –
163436311-SFB-910. V.V.S. also acknowledges support by the Russian Science
Foundation (project No. 22-72-00038).

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On $k$-free numbers over Beatty sequences

Wei Zhang

School of Mathematics and Statistics, Henan University, Kaifeng 475004, Henan,
China

<EMAIL_ADDRESS>
Abstract In this paper, we consider $k$-free numbers over Beatty sequences.
New results are given. In particular, for a fixed irrational number $\alpha>1$
of finite type $\tau<\infty$, any constant $\varepsilon>0$, we can show that

$\displaystyle\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{k}\end{subarray}}1-\frac{x}{\zeta(k)}\ll
x^{k/(2k-1)+\varepsilon}+x^{1-1/(\tau+1)+\varepsilon},$

where $\mathcal{Q}_{k}$ is the set of positive $k$-free integers and the
implied constant depends only on $\alpha,$ $\varepsilon,$ $k$ and $\beta.$
This improves previous results. The main new ingredient of our idea is an
employing of the double exponential sums of the type

$\sum_{1\leq h\leq H}\sum_{\begin{subarray}{c}1\leq n\leq x\\\
n\in\mathcal{Q}_{k}\end{subarray}}e(\vartheta hn).$

Keywords $k$-free numbers, exponential sums, Beatty sequence

2000 Mathematics Subject Classification 11L07, 11B83

##  1\. Introduction

In this paper, we are interested in $k-$free integers over Beatty sequences.
The so-called Beatty sequence of integers are defined by
$\mathcal{B}_{\alpha,\beta}:=\\{[\alpha n+\beta]\\}_{n=1}^{\infty},$ where
$\alpha$ and $\beta$ are fixed real numbers and $[x]$ denotes the greatest
integer not larger than $x$. The analytic properties of such sequences have
been studied by many experts. For example, one can refer to [1, 2, 3] and the
references therein. A number $q$ is called square-free integer if and only if
$m^{k}|q\Longrightarrow m=1.$ For sufficiently large $x\geq 1,$ it is well
known that

$\sum_{n\in\mathcal{Q}_{k}}n^{-s}=\frac{\zeta(s)}{\zeta(ks)},\ \ \ \sigma>1$

and

$\displaystyle\sum_{n\leq x,\
n\in\mathcal{Q}_{k}}1=\frac{x}{\zeta(k)}+O(x^{1/k}),$ (1.1)

where $\mathcal{Q}_{k}$ is the set of positive $k$-free integers. In this
paper, we are interested in the sum

$\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{k}\end{subarray}}1.$

In fact, this problem has been considered by many experts. For example, in
2008, Güloğlu and Nevans [7] proved that

$\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{2}\end{subarray}}1=\frac{x}{\zeta(2)}+O\left(\frac{x\log\log
x}{\log x}\right),$

where $\alpha>1$ is the irrational number of finite type.

Now we will recall some notion related to the type of $\alpha.$ The definition
of an irrational number of constant type can be cited as follows. For an
irrational number $\alpha,$ we define its type $\tau$ by the relation

$\tau:=\sup\left\\{\theta\in\mathbb{R}:\
\liminf_{\begin{subarray}{c}q\rightarrow\infty\\\
q\in\mathbb{Z}^{+}\end{subarray}}q^{\theta}\parallel\alpha
q\parallel=0\right\\}.$

Let $\psi$ be a non-decreasing positive function that defined for integers.
The irrational number $\alpha$ is said to be of type $<\psi$ if $q\parallel
q\alpha\parallel\geq 1/\psi(q)$ holds for every positive integers $q.$ If
$\psi$ is a constant function, then an irrational $\alpha$ is also called a
constant type (finite type). This relation between these two definitions is
that an individual number $\alpha$ is of type $\tau$ if and only if for every
constant $\tau$, there is a constant $c(\tau,\alpha)$ such that $\alpha$ is of
type $\tau$ with $q\parallel q\alpha\parallel\geq
c(\tau,\alpha)q^{-\tau-\varepsilon+1}.$

Recently, in [6], it is proved that

$\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{2}\end{subarray}}1=\frac{x}{\zeta(2)}+O\left(Ax^{5/6}(\log
x)^{5}\right),$

where $A=\max\\{\tau(m),1\leq m\leq x^{2}\\},$ and $\alpha>1$ is fixed
irrational algebraic number. More recently, Kim, Srichan and Mavecha [9]
improved the above result by showing that

$\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{k}\end{subarray}}1=\frac{x}{\zeta(k)}+O\left(x^{(k+1)/2k}(\log
x)^{3}\right).$

Recently, with some much more generalized arithmetic functions, in [1, 11],
one may also get some other estimates for such type sums. However, the
estimates of [1, 11] cannot be applied to an individual $\alpha.$ In this
paper, we can give the following formula.

######  Theorem 1.1.

Let $\alpha>1$ be a fixed irrational number of finite type $\tau<\infty$. Then
for any constant $\varepsilon>0$, we have

$\displaystyle\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{k}\end{subarray}}1-\alpha^{-1}\sum_{\begin{subarray}{c}1\leq
n\leq[\alpha x+\beta]\\\ n\in\mathcal{Q}_{k}\end{subarray}}1\ll
x^{k/(2k-1)+\varepsilon}+x^{1-1/(\tau+1)+\varepsilon},$

where the implied constant depends only on $\alpha,$ $\varepsilon,$ $k$ and
$\beta.$

Then by (1.1) and the above theorem, we can obtain the follows.

######  Corollary 1.2.

Let $\alpha>1$ be a fixed irrational number of finite type $\tau<\infty$. Then
for any constant $\varepsilon>0$, we have

$\displaystyle\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{k}\end{subarray}}1-\frac{x}{\zeta(k)}\ll
x^{k/(2k-1)+\varepsilon}+x^{1-1/(\tau+1)+\varepsilon},$

where the implied constant depends only on $\alpha,$ $\varepsilon,$ $k$ and
$\beta.$

In fact, our result relies heavily on the following double sum.

######  Theorem 1.3.

Suppose for some positive integers $a,q,h,$ $q\leq x,$ $h\leq H\ll x,$
$(a,q)=1$ and

$\displaystyle\left|\vartheta-\frac{a}{q}\right|\leq\frac{1}{q^{2}},$ (1.2)

then for sufficiently large $x$ and any $\varepsilon>0,$ we have

$\sum_{1\leq h\leq H}\sum_{\begin{subarray}{c}1\leq n\leq x\\\
n\in\mathcal{Q}_{k}\end{subarray}}e(\vartheta
hn)\ll\left(Hx^{k/(2k-1)}+q+Hx/q\right)x^{\varepsilon},$

where the implied constant may depend on $k$ and $\varepsilon.$

######  Remark 1.

One can also compare this result with the results of Brüdern-Perelli [4] and
Tolev [12]. By using the argument of [4, 12], one may get some better results
for some special cases.

##  2\. Proof of Theorem 1.1

We will start the proof by introducing some necessary lemmas.

######  Lemma 2.1.

Let $\alpha>1$ be of finite type $\tau<\infty$ and let $K$ be sufficiently
large. For an integer $w\geq 1,$ there exists $a,q\in\mathbb{N},$
$a/q\in\mathbb{Q}$ with $(a,q)=1$ and $q$ satisfying
$K^{1/\tau-\varepsilon}w^{-1}<q\leq K$ such that

$\left|\alpha w-\frac{a}{q}\right|\leq\frac{1}{qK}.$

###### Proof.

By Dirichlet approximation theorem, there is a rational number $a/q$ with
$(a,q)=1$ and $q\leq K$ such that $\left|\alpha w-a/q\right|<1/qK.$ Then we
have $\parallel qw\alpha\parallel\leq 1/K.$ Since $\alpha$ is of type
$\tau<\infty,$ for sufficiently large $K,$ we have $\parallel
qw\alpha\parallel\geq(qw)^{-\tau-\varepsilon}.$ Then we have $1/K\geq\parallel
qw\alpha\parallel\geq(qw)^{-\tau-\varepsilon}.$ This gives that $q\geq
K^{1/\tau-\varepsilon}w^{-1}.$ ∎

In order to prove the theorem, we need the definition of the discrepancy.
Suppose that we are given a sequence $u_{m},$ $m=1,2,\cdots$, $M$ of points of
$\mathbb{R}/\mathbb{Z}.$ Then the discrepancy $D(M)$ of the sequence is

$\displaystyle
D(M)=\sup_{\mathcal{I}\in[0,1)}\left|\frac{\mathcal{V}(\mathcal{I},M)}{M}-|\mathcal{I}|\right|,$
(2.1)

where the supremum is taken over all subintervals $\mathcal{I}=(c,d)$ of the
interval $[0,1),$ $\mathcal{V}(\mathcal{I},M)$ is the number of positive
integers $m\leq M$ such that $a_{m}\in\mathcal{I}$, and $|\mathcal{I}|=d-c$ is
the length of $|\mathcal{I}|.$

Without lose generality, let $D_{\alpha,\beta}(M)$ denote the discrepancy of
the sequence $\\{\alpha m+\beta\\},$ $m=1,2,\cdots$, $M$, where
$\\{x\\}=x-[x].$ The following lemma is from [2].

######  Lemma 2.2.

Let $\alpha>1.$ An integer $m$ has the form $m=[\alpha n+\beta]$ for some
integer $n$ if and only if

$0<\\{\alpha^{-1}(m-\beta+1)\\}\leq\alpha^{-1}.$

The value of $n$ is determined uniquely by $m.$

######  Lemma 2.3 (See Theorem 3.2 of Chapter 2 in [10]).

Let $\alpha$ be a fixed irrational number of type $\tau<\infty.$ Then, for all
$\beta\in\mathbb{R},$ we have

$D_{\alpha,\beta}(M)\leq M^{-1/\tau+o(1)},\ \ (M\rightarrow\infty),$

where the function implied by $o(1)$ depends only on $\alpha.$

######  Lemma 2.4 (see page 32 of [13]).

For any $\Delta\in\mathbb{R}$ such that $0<\Delta<1/8$ and $\Delta\leq
1/2\min\\{\gamma,1-\gamma\\},$ there exists a periodic function
$\Psi_{\Delta}(x)$ of period 1 satisfying the following properties:

  * •

$0\leq$$\Psi_{\Delta}(x)$$\leq 1$ for all $x\in\mathbb{R};$

  * •

$\Psi_{\Delta}(x)=\Psi(x)$ if $\Delta\leq x\leq\gamma-\Delta$ or
$\gamma+\Delta\leq x\leq 1-\Delta;$

  * •

$\Psi_{\Delta}(x)$ can be represented as a Fourier series

$\Psi_{\Delta}(x)=\gamma+\sum_{j=1}^{\infty}g_{j}e(jx)+h_{j}e(-jx),$

where

$\displaystyle\Psi(x)=\begin{cases}1\ \ &\textup{if}\ 0<x\leq\gamma,\\\ 0\ \
&\textup{if}\ \gamma<x\leq 1,\end{cases}$

and the coefficients $g_{j}$ and $h_{j}$ satisfy the upper bound

$\max\\{|g_{j}|,|h_{j}|\\}\ll\min\\{j^{-1},j^{-2}\Delta^{-1}\\},\ \ (j\geq
1).$

Suppose that $\alpha>1.$ Then we have that $\alpha$ and $\gamma=\alpha^{-1}$
are of the same type. This means that $\tau(\alpha)=\tau(\gamma)$ (see page
133 in [3]). Let $\delta=\alpha^{-1}(1-\beta)$ and $M=[\alpha x+\beta].$ Then
by Lemma 2.2, we have

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}1\leq n\leq x\\\
n\in\mathcal{Q}_{k}\end{subarray}}1&=\sum_{\begin{subarray}{c}1\leq m\leq M\\\
0<\\{\gamma m+\delta\\}\leq\gamma\\\
m\in\mathcal{Q}_{k}\end{subarray}}1+O(1)\\\ &=\sum_{\begin{subarray}{c}1\leq
m\leq M\\\ m\in\mathcal{Q}_{k}\end{subarray}}\Psi(\gamma
m+\delta)+O(1),\end{split}$ (2.2)

where $\Psi(x)$ is the periodic function with period one for which

$\displaystyle\Psi(x)=\begin{cases}1\ \ &\textup{if}\ 0<x\leq\gamma,\\\ 0\ \
&\textup{if}\ \gamma<x\leq 1.\end{cases}$

By a classical result of Vinogradov (see Lemma 2.4), it is known that for any
$\Delta$ such that $0<\Delta<1/8\ \textup{and}\
\Delta\leq\min\\{\gamma,1-\gamma\\}/2,$ there is a real-valued function
$\Psi_{\Delta}(x)$ satisfy the conditions of Lemma 2.4. Hence, by (2.2), we
can obtain that

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}1\leq n\leq x\\\
n\in\mathcal{Q}_{k}\end{subarray}}1&=\sum_{\begin{subarray}{c}1\leq m\leq M\\\
m\in\mathcal{Q}_{k}\end{subarray}}\Psi(\gamma m+\delta)+O(1)\\\
&=\sum_{\begin{subarray}{c}1\leq m\leq M\\\
m\in\mathcal{Q}_{k}\end{subarray}}\Psi_{\Delta}(\gamma
m+\delta)+O(1+V(I,M)M^{\varepsilon}),\end{split}$ (2.3)

where $V(I,M)$ denotes the number of positive integers $m\leq M$ such that

$\\{\gamma m+\delta\\}\in
I=[0,\Delta)\cup(\gamma-\Delta,\gamma+\Delta)\cup(1-\Delta,1).$

Since $|I|\ll\Delta,$ it follows from the definition (2.1) and Lemma 2.3 that

$\displaystyle V(I,M)\ll\Delta x+x^{1-1/\tau+\varepsilon},$ (2.4)

where the implied constant depends only on $\alpha.$ By Fourier expansion for
$\Psi_{\Delta}(\gamma m+\delta)$ (Lemma 2.4) and changing the order of
summation, we have

$\displaystyle\begin{split}\sum_{\begin{subarray}{c}1\leq m\leq M\\\
m\in\mathcal{Q}_{k}\end{subarray}}&\Psi_{\Delta}(\gamma m+\delta)\\\
&=\gamma\sum_{\begin{subarray}{c}m\leq M\\\
m\in\mathcal{Q}_{k}\end{subarray}}1+\sum_{k=1}^{\infty}g_{k}e(\delta
k)\sum_{\begin{subarray}{c}1\leq m\leq M\\\
m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km)\\\
&+\sum_{k=1}^{\infty}h_{k}e(-\delta k)\sum_{\begin{subarray}{c}1\leq m\leq
M\\\ m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km).\end{split}$ (2.5)

By Theorem 1.3, Lemma 2.1 and Lemma 2.4, we see that for $0<k\ll
x^{(4k-4)/(2k-1)+\varepsilon}$, we have

$\displaystyle\begin{split}\sum_{1\leq k\leq
x^{(4k-4)/(2k-1)+\varepsilon}}g_{k}e(\delta k)\sum_{\begin{subarray}{c}1\leq
m\leq M\\\ m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km)\ll
x^{k/(2k-1)+\varepsilon}+x^{1-1/(\tau+1)+\varepsilon},\end{split}$ (2.6)

where we have also used the fact that $\alpha$ and $\alpha^{-1}$ are of the
same type (finite type). Similarly, we have

$\displaystyle\begin{split}\sum_{1\leq k\leq
x^{(4k-4)/(2k-1)+\varepsilon}}h_{k}e(-\delta k)\sum_{\begin{subarray}{c}1\leq
m\leq M\\\ m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km)\ll
x^{k/(2k-1)+\varepsilon}+x^{1-1/(\tau+1)+\varepsilon},\end{split}$ (2.7)

On the other hand, the trivial bound

$\sum_{\begin{subarray}{c}1\leq m\leq M\\\
m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km)\ll x$

implies that

$\displaystyle\begin{split}\sum_{k\geq
x^{(4k-4)/(2k-1)+\varepsilon}}g_{k}e(\delta k)\sum_{\begin{subarray}{c}1\leq
m\leq M\\\ m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km)&\ll
x^{1+\varepsilon}\sum_{k\geq
x^{(4k-4)/(2k-1)+\varepsilon}}k^{-2}\Delta^{-1}\\\ &\ll
x^{k/(2k-1)+\varepsilon}\end{split}$ (2.8)

and

$\displaystyle\begin{split}\sum_{k\geq
x^{(4k-4)/(2k-1)+\varepsilon}}h_{k}e(-\delta k)\sum_{\begin{subarray}{c}1\leq
m\leq M\\\ m\in\mathcal{Q}_{k}\end{subarray}}e(\gamma km)&\ll
x^{1+\varepsilon}\sum_{k\geq
x^{(4k-4)/(2k-1)+\varepsilon}}k^{-2}\Delta^{-1}\\\ &\ll
x^{k/(2k-1)+\varepsilon}\end{split}$ (2.9)

where $\Delta=x^{-(k-1)/(2k-1)+\varepsilon}.$ Inserting the bounds (2.6)-(2.9)
into (2.5), we have

$\displaystyle\sum_{\begin{subarray}{c}1\leq n\leq x\\\ [\alpha
n+\beta]\in\mathcal{Q}_{k}\end{subarray}}1-\alpha^{-1}\sum_{\begin{subarray}{c}1\leq
n\leq[\alpha x+\beta]\\\ n\in\mathcal{Q}_{k}\end{subarray}}1\ll
x^{k/(2k-1)+\varepsilon}+x^{1-1/(\tau+1)+\varepsilon},$

where the implied constant depends on $\alpha,$ $k,$ $\beta$ and
$\varepsilon.$ Substituting this bounds and (2.4) into (2.3) and choosing
$\Delta=x^{-(k-1)/(2k-1)+\varepsilon}$, we complete the proof of Theorem 1.1.

##  3\. Proof of Theorem 1.3

By Dirichlet hyperbolic method, we have

$\displaystyle\sum_{1\leq h\leq H}\sum_{\begin{subarray}{c}1\leq n\leq x\\\
n\in\mathcal{Q}_{k}\end{subarray}}1$ $\displaystyle=\sum_{1\leq h\leq
H}\sum_{1\leq m^{k}\leq y}\mu(m)\sum_{1\leq l\leq x/m^{k}}e(\alpha m^{k}lh)$
$\displaystyle+\sum_{1\leq h\leq H}\sum_{1\leq l\leq x/y}\sum_{1\leq m^{k}\leq
x/l}\mu(m)e(\alpha m^{k}lh)$ $\displaystyle-\sum_{1\leq h\leq H}\sum_{1\leq
m^{k}\leq y}\mu(m)\sum_{1\leq l\leq x/y}e(\alpha m^{k}lh),$

where $y$ is certain parameter to be chosen later. By the well known estimate

$\sum_{1\leq n\leq x}e(n\alpha)\leq\min\left(x,\frac{1}{2||\alpha||}\right),$

we have

$\displaystyle\sum_{1\leq h\leq H}\sum_{1\leq m^{k}\leq y}\mu(m)\sum_{1\leq
l\leq x/m^{k}}e(\alpha m^{k}lh)$ $\displaystyle\ll\sum_{1\leq h\leq
H}\sum_{1\leq m^{k}\leq y}\min\left(x/m^{k},\frac{1}{2||\alpha
hm^{k}||}\right)$ $\displaystyle\ll\sum_{1\leq h\leq H}\sum_{1\leq m\leq
y}\min\left(x/m,\frac{1}{2||\alpha hm||}\right)$
$\displaystyle\ll(Hy)^{\varepsilon}\sum_{1\leq n\leq
Hy}\min\left(Hx/n,\frac{1}{2||\alpha n||}\right)$
$\displaystyle\ll(Hy)^{\varepsilon}\left(Hy+Hx/q+q\right),$

where we have used the following lemma.

######  Lemma 3.1 (See section 13.5 in [8]).

For

$\left|\theta-a/q\right|\leq q^{-2},$

$a,q\in\mathbb{N}$ and $(a,q)=1,$ then we have

$\sum_{1\leq n\leq M}\min\left\\{\frac{x}{n},\frac{1}{2\parallel
n\theta\parallel}\right\\}\ll\left(M+q+xq^{-1}\right)\log 2qx,$

where $\parallel u\theta\parallel$ denotes the distance of $u$ from the
nearest integer.

For the second the sum, we can use the exponential sum for Möbius function. We
have

$\sum_{1\leq h\leq H}\sum_{1\leq l\leq x/y}\sum_{1\leq m^{k}\leq
x/l}\mu(m)e(\alpha m^{k}lh)\ll Hxy^{1-1/k}(\log x)^{-A},$

where $A$ is any positive constant. For the third sum we need the following
lemma.

######  Lemma 3.2 (See section 13.5 in [8]).

For

$\left|\theta-a/q\right|\leq q^{-2},$

$a,q\in\mathbb{N}$ and $(a,q)=1,$ then we have

$\sum_{1\leq n\leq M}\min\left\\{x,\frac{1}{2\parallel
n\theta\parallel}\right\\}\ll\left(M+x+Mx/q+q\right)\log 2qx,$

where $\parallel u\theta\parallel$ denotes the distance of $u$ from the
nearest integer.

By Lemma 3.2, we have

$\displaystyle\sum_{1\leq h\leq H}\sum_{1\leq m^{k}\leq y}\mu(m)\sum_{1\leq
l\leq x/y}e(\alpha m^{k}lh)$ $\displaystyle\ll\sum_{1\leq h\leq H}\sum_{1\leq
m^{k}\leq y}\min\left(x/y,\frac{1}{2||\alpha hm^{k}||}\right)$
$\displaystyle\ll\sum_{1\leq h\leq H}\sum_{1\leq m\leq
y}\min\left(x/y,\frac{1}{2||\alpha hm||}\right)$
$\displaystyle\ll(Hy)^{\varepsilon}\sum_{1\leq n\leq
Hy}\min\left(x/y,\frac{1}{2||\alpha n||}\right)$
$\displaystyle\ll(Hy)^{\varepsilon}\left(Hy+x/y+Hx/q+q\right).$

Choosing $y=x^{k/(2k-1)}$ completes the proof of Theorem 1.3.

$\mathbf{Acknowledgements}$ I am deeply grateful to the referee(s) for
carefully reading the manuscript and making useful suggestions.

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al., 2003], and $g\simeq 1.62$ m2s-1 at the Moon surface, $H\simeq 1700$ km
close to the Moon radius, and $\Delta T$ in the range $500-1000$ K [Solomatov,
2000], we obtain $Ra\simeq 10^{28}-10^{29}$ and $F$ in the range $2\times
10^{5}-5\times 10^{5}$ J m${}^{-2}~{}$s-1.

The velocity for turbulent convection scales as [Kraichnan, 1962]

$U\simeq 0.6\left(\frac{\alpha gFH}{\rho C_{p}}\right)^{1/3},$ (F3)

where $F$ is the heat flux from the ocean to the atmosphere, $C_{p}$ the
thermal capacity of liquid silicates, $\rho$ the magma ocean density. Using
the above estimate for $F$, and assuming $\rho\simeq 3340$ kg m-3 for the Moon
and $C_{p}\simeq 10^{3}$ J kg${}^{-1}~{}$K-1 [Solomatov, 2000], we find
convective velocities $U\simeq 1-1.5$ m s-1. The associated turnover time
scale in the magma ocean is $t_{turn}=H/U\simeq 14-20$ days. This implies that
the silicates below the Moon surface are renewed in about $14-20$ days.

##  Appendix G Silicate volume brought to the surface

Assuming volatiles exsolve from the magma ocean as gas bubbles [Hamano et al.,
2013, Salvador et al., 2017], the volume of liquid silicates that can interact
with the atmosphere, before the magma ocean solidifies at time $t_{sol}$, is

$V_{\rm sil}\simeq 4\pi R_{\rm m}^{2}Ut_{\rm sol}\simeq 4\pi R_{\rm
m}^{3}\frac{t_{\rm sol}}{t_{\rm turn}}$ (G1)

where $R_{\rm m}$ is the Moon radius. Assuming a solidification time ranges
from $200$ years[Tang and Young, 2020] to $1000$ yrs[Solomatov, 2000], we
obtain

$4\times 10^{3}V_{\rm moon}<V_{\rm sil}<8\times 10^{4}V_{\rm moon},$ (G2)

where $V_{\rm moon}$ is the Moon volume. In this case, $V_{\rm sil}$ is orders
of magnitude larger than the volume of the Moon. Hence, the whole magma ocean
has time to interact with the atmosphere. We therefore expect that including
the dynamics of the magma ocean will not affect the leading-order conclusions
of this paper.

##  Appendix H Location of the photosphere

We have computed the atmospheric optical depth $\tau(r)$ assuming that liquid
droplets are the only source of opacity. The droplet radius, $r_{\rm d}$ is a
fixed parameter and the droplets density is $\rho_{\rm d}$ . With these
assumptions the optical depth, integrated from the Hill Sphere down to some
distance from the Moon’s center, r, is :

$\tau(r)=\frac{3}{4\pi\rho_{\rm d}r_{\rm d}}\int_{r}^{R_{g}}\rho(r)X(r)dr$
(H1)

where $R_{\rm h}$ is the Hill sphere location, $\rho$ is the density of the
liquid-gas mixture and $X$ is the droplet mass fraction. We define the
photosphere $Z_{\rm phot}$ as the location where $\tau(Z_{\rm phot})=1$. We
take $\rho_{\rm d}=3200{\rm kg/m}^{3}$. Results are presented for different
Earth-Moon distances and for different droplet radii : 0.01 mm (Figure 26) and
1 mm (Figure 27). As expected, for larger particle size the photosphere is
closer to the ground (i.e. the atmosphere is more transparent), and for a
larger Earth-Moon distance also. Interestingly the temperature is the main
parameter controlling the location of the photosphere. We find that the
transition from an opaque atmosphere (with $Z_{\rm phot}\simeq R_{\rm h}$) to
a transparent atmosphere ( $Z_{\rm phot}\simeq R_{\rm m}$) occurs for
temperature between 1800 K and 2400 K for an Earth-Moon distance =
$5R_{\oplus}$.

Figure 26: Location of the photosphere (color-coded) as a function of the
surface temperature and the liquid mass fraction, $X$, at the surface. The
color codes the relative distance of the photosphere compared to the surface
or the Hill radius: $(Z_{\rm phot}-R_{\rm m})/(R_{\rm h}-R_{\rm m})$. Yellow
color means that the photosphere is at the Hill radius, black color means that
the photosphere is at the ground-level (i.e the atmosphere is transparent).
Droplet radius is $10^{-5}$ m. The 4 panels correspond to different Earth-Moon
distances $A_{\rm m}$. White zone correspond to regions where no escape
occurs. Figure 27: Location of the photosphere (color-coded) as a function of
the surface temperature and the liquid mass fraction, X, at the surface. Same
legend as above. The droplet radius is $10^{-3}$ m.

##  Appendix I Composition of the vapor atmosphere

The saturated vapor composition of an atmosphere at equilibrium with a magma
with BSE composition was recently published [Visscher and Fegley, 2013].
However this work does not extend below 2000 K, a temperature regime of
interest to explain the Moon composition. Using up to date thermodynamic
data(Sossi and Fegley Jr. [2018]), we have computed the vapor composition for
$T<2000$ K. The equilibrium partial pressure may be calculated for any given
vaporisation reaction:

$\left(M^{x+n}O_{\frac{x+n}{2}}\right)_{l}=\left(M^{x}O_{\frac{x}{2}}\right)_{g}+\frac{n}{4}O_{2}$
(I1)

where $l$ denotes the liquid phase, $g$ the gas phase, $x$ the oxidation state
of the metal in the gas phase and $n$ the coefficient of stoichiometry (an
integer value). We note $K$ the equilibrium constant of this reaction. From
Equation I1 the equilibrium partial vapor pressure of species $M^{x}O_{x/2}$
(designated as species $i$ for short) is obtained as follows :

$p\left(M^{x}O_{\frac{x}{2}}\right)=\frac{KX(i)\gamma(i)}{f(O_{2})^{n/4}}$
(I2)

where $X(i)$ is the mole fraction of species $i$ (see table below),
$\gamma(i)$ its activity coefficient and $f(O_{2})$ the oxygen fugacity.

Here we estimate the activity coefficients of Na, K, Cu and Zn based on
thermodynamic measurements from laboratory experiments on basaltic melts (Na,
De Maria et al. 1971, Mathieu et al. 2008 and Sossi et al. 2019; K, De Maria
et al. 1971, Hastie et al. 1983 and Sossi et al. 2019; Cu, Sossi et al. 2019,
and Zn, Reyes and Gaskell, 1983 and Sossi et al. 2019). The ratio
$\gamma(Na)$/$\gamma(K)$ is set to remain constant as a function of
temperature (cf. Charles, 1967) where $\gamma(Na)$ is taken to be $10^{-3}$ at
1700 K and $\gamma(K)=5\times 10^{-4}$ at the same temperature. Uncertainties
in these values are as large as a factor of 5, see also Jiang et al. (2019).
For example, Mathieu et al. 2008 provide a value of $\gamma(Na)$ = $10^{-3}$
at 1623 K, Sossi et al. (2019) report the same value at 1673 K, whereas the
Knudsen Effusion Mass Spectrometry measurements of De Maria et al. (1971)
suggest a $\gamma(Na)$ of $5\times 10^{-3}$ at 1500 K. Copper may exist in its
2+ and 1+ oxidation states in silicate melts (Holzheid and Lodders, 2001),
however at the oxygen fugacities produced by evaporation of the lunar silicate
( FMQ), it occurs uniquely as $Cu^{+}$. We assume that the major element
oxides ($MgO$, $FeO$ and $SiO_{2}$) dissolve ideally into silicate melts (e.g.
O’Neill and Eggins, 2002; Beckett, 2002).

Here, we take the model of Visscher and Fegley (2013) as an input parameter to
calculate $f(O_{2})$. This model compares well with measurements of oxygen
fugacities above vaporising silicates above 1400 K (see Fig. 4 of ref. Sossi
and Fegley Jr. [2018]). The values of $n$, the stoichiometric coefficient of
the vaporisation reaction, is an integer value equal to 1 (Na, K) and 2 (Mg,
Fe, Si, Zn). As such, the condensed species considered are: $NaO_{0.5}$,
$KO_{0.5}$, $CuO_{0.5}$ $MgO$, $FeO$, $SiO_{2}$, and $ZnO$, all (l) and the
gas species are: $Na^{0}$, $K^{0}$, $Cu^{0}$, $Mg^{0}$, $Fe^{0}$, $SiO$, and
$Zn^{0}$. We consider only these eight elements as they have the most abundant
species in the vapour ¡2000 K [Visscher and Fegley, 2013].

Following our experimental data of BSE vaporisation we use the values for
$f(O_{2})$ reported in Table 4.

Temperature (K) | 1500 | 1600 | 1700 | 1800 | 1900 | 2000
---|---|---|---|---|---|---
$f(O_{2})$ (bar) | $2.04\times 10^{-7}$ | $5.79\times 10^{-7}$ | $1.53\times 10^{-6}$ | $3.82\times 10^{-6}$ | $9.03\times 10^{-6}$ | $2.04\times 10^{-5}$
Table 4: The $f(O_{2})$ of an atmosphere in equilibrium with a magma ocean of
BSE composition. Oxygen fugacities are close to the FMQ buffer, as attested to
by measurements above natural basalts (De Maria et al. 1971) and olivine
(Costa et al. 2017), and as modelled by Visscher and Fegley (2013). Here we
use the values modelled by Visscher and Fegley (2013) for a BSE composition.

Here are the thermodynamic parameters for the different species of interest.
The molar abundances :

Species | X (Mole Fraction)
---|---
Na | $5.93\times 10^{-3}$
K | $3.5\times 10^{-4}$
Cu | $2.6\times 10^{-5}$
Zn | $4.95\times 10^{-5}$
Mg | $0.2217$
Fe | $0.063$
Si | $0.2122$
Table 5: Composition of BSE used in this paper.

In order to predict the temperature dependence of the activity coefficients of
Na, K and Zn in silicate melts, we draw upon pre-existing experimental data.
Charles (1967) determined the activity coefficients of $Na_{2}O$ and $K_{2}O$
in alkali oxide - $SiO_{2}$ binary systems by application of the cryoscopic
equation to pre-existing vaporisation data, thereby determining heats of
solution of oxide components in the silicate melt. The two quantities are
related by the van’t Hoff equation:

$\frac{dln{\gamma(i)}}{d(1/T)}=\frac{\Delta H_{sol}}{R}$ (I3)

We take values of $\Delta H_{sol}$ from Charles, 1967 for $NaO_{0.5}$ (-112
kJ/mol) and fix $\gamma(KO_{0.5})$ as $0.5\times\gamma(NaO_{0.5})$, while we
use the values determined by Reyes and Gaskell (1983), -135 kJ/mol and Sossi
et al. (2019), -143 kJ/mol for ZnO, with 121 kJ/mol for $CuO_{0.5}$ from Sossi
et al. (2019). All quantities are extracted presuming infinite dilution of the
trace component in the silicate melt.

The estimated activity coefficients are therefore the following :

Temperature (K) | 1500 | 1600 | 1700 | 1800 | 1900 | 2000
---|---|---|---|---|---|---
$\gamma$ for $NaO_{0.5}$ | 0.0004 | 0.0008 | 0.001 | 0.002 | 0.003 | 0.004
$\gamma$ for $KO_{0.5}$ | 0.0002 | 0.0004 | 0.0005 | 0.001 | 0.0015 | 0.002
$\gamma$ for $CuO_{0.5}$ | 37.6 | 20.5 | 12 | 7.5 | 4.9 | 3.3
$\gamma$ for $ZnO$ | 0.18 | 0.42 | 0.56 | 0.65 | 0.71 | 0.75
Table 6: Activity coefficients used in this paper.

The equilibrium constant may be calculated using thermodynamic data (e.g.
Chase 1998) using the expression:

$\Delta G=-RTln(K)$ (I4)

in which

$\Delta G=\Delta H-T\Delta S$ (I5)

We use the following enthalpies and entropies :

  * •

$\Delta G(Na)=260720-T\times 118.09$ J/mol (Lamoreaux and Hildenbrand, 1984)

  * •

$\Delta G(K)=210056-T\times 101.71$ J/mol(Lamoreaux and Hildenbrand, 1984)

  * •

$\Delta G(Zn)=405910-T\times 174.4$ J/mol (Lamoreaux et al., 1987)

  * •

$\Delta G(Cu)=377700-T\times 135.0$ J/mol (O’Neill and Pownceby, 1993)

  * •

$\Delta G(Mg)=662155-T\times 184.01$ J/mol (Chase, 1998)

  * •

$\Delta G(Fe)=636430-T\times 178.46$ J/mol (Chase, 1998)

  * •

$\Delta G(Si)=778440-T\times 242.29$ J/mol (Chase, 1998)

The resulting partial pressure are plotted in Figure 14.

##  Appendix J Caveats and approximations

This paper aims to show that degassing during the lunar assembling is
potentially an important process, and may explain the depletion of lunar
material in moderately volatile elements. Of course, to make the computation
tractable, several approximations must be done, and before making a fully
self-consistent model we aim here to show that first-orders calculation show
that tidally driven evaporation is a viable process. Our simplifications are
summarized below:

  * •

Radiative heating and cooling : The present work does not consider radiative
heating and cooling processes that may act during the gas expansion. Only
adiabatic cooling is considered. This would require to couple the dynamical
model to radiative transfer code, that has never been developed for an
atmosphere expanding above a magma ocean. Computation of the radiative
diffusion timescale following Tremblin et al. (equation 17) shows that in the
optically thick case the radiative diffusion timescale is shorter than the
expansion timescale (a few 10s compared to a few $10^{4}$s for droplets in the
range of micrometer size), assuming the source of opacity are the droplets in
the flow. This implies that the thermal radiation may potentially modify the
temperature during the expansion if important temperature gradient is present.
In the absence of a full radiative transfer model, we compare two length
scales : the temperature scale height ($L_{\rm temp}$):

$L_{\rm temp}=\frac{T}{\partial T/\partial r}$ (J1)

and the photon mean free path ($L_{\rm rad})$:

$L_{\rm rad}=\frac{1}{\kappa\rho}$ (J2)

where $\kappa$ is the local opacity (computed with assuming that the opacity
source is the presence or droplets with a fixed radius). If $L_{\rm
temp}>>L_{\rm rad}$, this means that a photon experiences several scattering
events before reaching a region with significantly different temperature. The
assumption of local thermal equilibrium (LTE) is therefore valid, and we do
not need to consider the energy loss due to radiation. In the opposite case,
the photons can escape without being absorbed and radiative cooling effect
should not be neglected. We plot in Figure 28 the ratio between the two length
scales for different atmospheric profiles, $L_{\rm temp}$ and $L_{\rm rad}$
being evaluated at the sonic point. We find that at high temperature $L_{\rm
temp}>L_{\rm rad}$ and at low temperature $L_{\rm temp}<L_{\rm rad}$, the
transition between the two regimes situating at $T=1400$ K for 1 micron radius
droplets and $T=1700$ K for 1 mm radius droplets. We conclude that above such
temperature, the present work neglecting radiative cooling effects stays
valid, whereas at lower temperature complete radiative modeling should be
considered.

Figure 28: $Log_{10}(L_{\rm temp}/L_{\rm rad})$ evaluated at the sonic point
for a Moon at 5 $R_{\oplus}$ as a function of the surface temperature and the
liquid mass fraction (X). Left : for liquid droplets size=1 micron; Right:for
liquid droplets size=1mm.

Taking into account radiative cooling and heating from the Moon surface would
require to develop a radiative transfer model, well beyond the scope of the
paper that intends to identify a new escaping process and to point out its
potential importance. However, for the simple case of a collisionless gas,
with no pressure and at thermal equilibrium with the hot radiating Moon’s
surface, we show in Figure 2.a that the gas temperature becomes always larger
than the necessary temperature for escaping for a Moon within 5$R_{\oplus}$ of
the Earth center’.

This shows that escape is an unavoidable process for a fully grown Moon.
Escape becomes even more intense for a Moon’s building block, with a fraction
of a lunar mass (Figure 2.b, and core text).

  * •

1D calculation : In this paper we have done the calculation in 1D, along the
line joining the center of the Earth and the center of the Moon. So, strictly
speaking, the computed flow would be only valid along this line, and it cannot
be assumed that the flow is emitted at $4\pi$ steradians on the surface. It is
why, for simplicity purpose we have assume that the flow is arbitrarily
emitted on a solid angle of 1 steradian. The full calculation would need a
full 3D treatment, as the gas will probably circulate around the Moon before
escaping. But such model is still beyond the state of the art, but could be
done in a close future.

##  Appendix K Lost mass fractions

Let $M^{M}$ and $M^{E}$ the Moon and Earth Mass respectively today. Let
$\mu_{i}^{E}$ the mass fraction of an element $i$ in the BSE, $\mu_{i}^{M}$
the mass fraction of an element in the Moon today, and $\mu_{i}^{v}$the mass
fraction of an element $i$ in the vapor at some temperature. We call $M^{L}$
the total mass lost during degassing of the Moon. $M^{L}=M^{L}_{v}+M^{L}_{l}$
where $M^{L}_{v}$ is the mass lost in the form of gas, and $M^{L}_{l}$ is the
mass lost in the form of droplets dragged by the gas. We call $M_{i}^{M}$ the
total mass of an element in the Moon today and $M_{i}^{E}$ the total mass of
an element in the BSE. We assume that the Moon is degassed at constant
temperature for simplicity so that $\mu_{i}^{v}$ is a constant for every
species. The Initial mass of the Moon is $M^{M}+M^{L}$ and final mass is
$M^{M}$. If we assume that the Moon material was initially BSE composition we
have :

$M_{i}^{M}=\mu_{I}^{E}(M^{M}+M^{L})-\mu_{i}^{v}M^{L}_{v}-\mu_{i}^{E}M^{L}_{l}$
(K1)

The first term is the initial mass of element $i$ the second term is the mass
lost in vapor form, and the third term is the mass lost in liquid form
(assumed to have BSE composition). Since $M^{L}_{l}=xM^{L}$ and
$M^{L}_{v}=(1-x)M^{L}$ ; we get

$M_{i}^{M}=\mu_{I}^{E}(M^{M}+M^{L}_{v})-\mu_{i}^{v}M^{L}_{v}$ (K2)

Dividing by $M^{M}$ , the Moon mass, we obtain

$\mu_{i}^{M}=\mu_{I}^{E}\frac{M^{M}+M^{L}_{v}}{M^{M}}-\mu_{i}^{v}\frac{M^{L}_{v}}{M^{M}}\Rightarrow$
(K3)
$\mu_{i}^{M}=\mu_{I}^{E}+\frac{M^{L}_{v}}{M^{M}}(\mu_{i}^{E}-\mu_{i}^{v})$
(K4)

In the above equation the effect of degassing on the mass fraction in the Moon
($\mu_{i}^{M}$) is obvious. No degassing means $M^{L}_{v}=0$ so $\mu_{i}^{M}$
stays equal to the BSE value $\mu_{I}^{E}$. if $\mu_{i}^{v}>\mu_{i}^{E}$
(which is the case for volatile species in general) then degassing lowers the
value of $\mu_{i}^{M}$.

From Equation K4 we derive the mass fraction of an element remaining in the
Moon, compared to its mass fraction in the BSE:

$\frac{\mu_{i}^{M}}{\mu_{i}^{E}}=1+\frac{M^{L}_{v}}{M^{M}}(1-\frac{\mu_{i}^{v}}{\mu_{i}^{E}})$
(K5)

From the above equation we see that fractionation, compared to the BSE, is
only possible through vapor loss. If $\mu_{i}^{v}<\mu_{i}^{E}$ then this
element will be enriched in the Moon, conversely elements with
$\mu_{i}^{v}>\mu_{i}^{E}$ will be depleted in the Moon.

##  Appendix L Enhancement of Volatile Loss due to XUV Heating from the Proto-
Earth

While our model treats tidally-driven escape heated uniquely from the bottom
by the magma ocean, additional heating from the top by XUV irradiation as at
exoplanets [Erkaev et al., 2007] and even Kuiper Belt Objects [Johnson et al.,
2015] gives further credence to the Moon’s devolatilization by atmospheric
escape. By employing the atmospheric evolution model DISHOOM [Gebek and Oza,
2020] we investigate mass loss due to surface and upper atmospheric heating.
We find that while the tenuous Jeans escape rates due to surface heating are
quite small, catastrophic volatile loss ($<100$ Myr) must still occur due to a
remarkably bright proto-earth[Saxena et al., 2017] heating the upper
atmosphere via XUV energy-limited escape. Figure 29 shows three scenarios of
three different estimates of the proto-Earth temperature with distance. We
adopt a heating efficiency of 0.35 similar to Io’s volcanic
atmosphere[Lellouch et al., 1992]. A conservative assumption, neglecting
atmospheric sputtering, that the only proto-earth irradiation driving escape
is due to the X-ray and UV component ($\sim 3.6\times 10^{-6}L_{\rm bol}$)
still yields a total mass loss of $\sim 0.1\%$ of a lunar mass, in 1 Myr. The
mass loss rates on the order of $\sim 10^{6}$ kg/s are then quite similar to
an early magma-ocean planet at 1 AU orbiting an M-dwarf[Bower et al., 2019] or
a close-in exomoon[Oza et al., 2019]. Considering the timescales and
proximities studied in this work, a catastrophic volatile loss scenario for
the Moon seems inevitable.

Figure 29: Energy-limited escape from the proto-Moon due to irradiation of a
molten proto-Earth at three different effective temperatures $T_{\oplus}$:
2500 K (red), 2450 K (black), 2300 K (blue).