# SCATTER: A New Common Envelope Formalism

Rosanne Di Stefano<EMAIL_ADDRESS>Harvard-Smithsonian Center for
Astrophysics, 60 Garden St, Cambridge, MA 02138, US  Matthias U. Kruckow
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China  Key
Laboratory for the Structure and Evolution of Celestial Objects, Chinese
Academy of Sciences, Kunming 650011, China  Département d’Astronomie,
Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
Gravitational Wave Science Center (GWSC), Université de Genève, CH-1211
Geneva, Switzerland  Yan Gao Yunnan Observatories, Chinese Academy of
Sciences, Kunming 650011, China  Key Laboratory for the Structure and
Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011,
China  University of Chinese Academy of Sciences  Institute of Gravitational
Wave Astronomy, School of Physics and Astronomy, University of Birmingham,
Birmingham, B15 2TT, UK  Patrick G. Neunteufel Max Planck Institut für
Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching bei München, Germany
Chiaki Kobayashi Chiaki Kobayashi<EMAIL_ADDRESS>Centre for
Astrophysics Research, Department of Physics, Astronomy and Mathematics,
University of Hertfordshire, Hatfield, AL10 9AB, UK

(Received December 12, 2022; Revised December 22, 2022; Accepted December 23,
2022)

###### Abstract

One of the most mysterious astrophysical states is the common envelope (CE)
phase of binary evolution, in which two stars are enshrouded by the envelope
shed by one of them. Interactions between the stars and the envelope shrinks
the orbit. The CE can lead to mergers or to a subsequent phase of
interactions. Mergers may involve any combination of two compact objects
and/or stars. Some involving white dwarfs, may produce Type Ia supernovae,
while merging neutron stars may yield gamma-ray bursts, and merging compact
objects of all kinds produce gravitational radiation. Since CEs can arise from
a variety of different initial conditions, and due to the complexity of the
processes involved, it is difficult to predict their end states. When many
systems are being considered, as in population synthesis calculations,
conservation principles are generally employed. Here we use angular momentum
in a new way to derive a simple expression for the final orbital separation.
This method provides advantages for the study of binaries and is particularly
well-suited to higher order multiples, now considered to be important in the
genesis of potential mergers. Here we focus on CEs in binaries, and the
follow-up paper extends our formalism to multiple star systems within which a
CE occurs.

††journal: ApJ

##  1 Introduction

The idea of the common envelope (CE) was suggested by Paczynski (1976), to
explain the existence of cataclysmic variables . These are close binaries,
typically with orbital periods smaller than about 9 hours, in which a low-mass
star transfers mass to a white dwarf (WD). Since the WD must have once been
the core of a star far larger than the separation between the WD and its
present-day companion, the CE was invoked to bring the WD and its companion
star together. There is a natural way for a CE to be generated. When a star
fills its Roche lobe (RL) but cannot transfer mass in a stable manner to its
companion, its envelope detaches on a dynamical time scale and comes to
surround both the core of the donor and its companion. The stars within the
envelope spiral closer to each other until the envelope is ejected. For a
recent review see Roepke & De Marco (2022).

Many physical processes, including gravitation, electromagnetism, and fluid
dynamics determine the final state of the system (Ivanova et al., 2013, 2020).
Advances in simulations are making it possible to explore individual cases in
some detail, an approach that already has begun to yield insight into the
evolution of the CE phase (e.g., Ricker & Taam, 2012; Passy et al., 2012; Sand
et al., 2020; Kramer et al., 2020). Hydrodynamical simulations can track the
mass in the envelope and incorporate a wealth of physical effects, but can
only be applied to a small number of individual systems, and rely on input
assumptions that introduce a level of systematic uncertainty that difficult to
quantify.

Much of the interest in the wider community focuses on the end states of the
CE phase. The CE can lead to mergers of the components during or after the
ejection of the envelope, yielding phenomena as diverse as Type Ia supernovae
(SNe Ia), accretion induced collapse, and double compact object mergers
involving neutron stars (NSs) and/or black holes (BHs). Some of these
processes are important for the origin of elements. Additionally, the end
state of the CE can also serve as the starting state for a future phase of
mass transfer.

It is therefore tempting proposition to come up with a model that is capable
of predicting the end state of a CE evolution phase from its initial
conditions, and many efforts have accordingly been directed towards doing so.
Among these efforts, some of the most notable have culminated in the $\alpha$
formalism (Webbink 1984, Iben & Livio 1993, see also Dewi & Tauris 2000) and
the $\gamma$ formalism (Nelemans et al. 2000, Nelemans & Tout 2005, see also
Soker 2004), two prescriptions which are widely used for this purpose (e.g.,
Toonen et al. 2012, Toonen & Nelemans 2013, Korol et al. 2022). The $\alpha$
formalism has its foundations in conservation of energy, pointing out that the
energy used to eject the envelope must have come from the energy donated by
the orbit of the binary pair as it shrinks. It asserts that, once the energy
conversion efficiency (taking into account the geometry of the system)
${\alpha}_{\rm CE}{\lambda}$ at which orbital energy is converted into
envelope internal energy is ascertained, the final state can be easily found.
The $\gamma$ formalism, on the other hand, is rooted in conservation of
angular momentum, arguing in a similar way that the angular momentum required
to expand the envelope must be sourced from the binary orbit. Thus, a given
fraction of angular momentum of the entire system must be lost in order to
eject a given common envelope which accounts for a given fraction of the total
mass of the system, and the ratio between the two fractions is $\gamma$.
However, both formalisms run into problems in certain situations, such as when
the $\alpha$ formalism encounters situations in which a CE leads to a net
increase in the binary separation, and neither is a perfect predictor for all
CE end states (see also De Marco et al., 2011; Hirai & Mandel, 2022).
Furthermore, neither prescription can be easily transferred to the study of
triple systems and higher-order multiples entirely engulfed by CEs, which are
becoming increasingly relevant to studies of stellar evolution today (e.g.,
Gao et al., 2023).

In this paper we present a new CE formalism which is naturally well suited to
triple systems and higher order multiples. In our formalism, each mass in the
system transfers angular momentum to material in the CE. Unlike the $\gamma$
formalism, our prescription pays attention to how much each individual
component contributes to the angular momentum being taken away, and in doing
so, renders the relevant free parameters calculable using first principles. We
do not assume anything about the mechanism that ejects the envelope. Instead
we focus entirely on the angular momentum transferred from or to the
individual stellar components of the multiple-star system. Our primary
assumption is that it is the transfer of angular momentum to and from the
individual stars that changes their orbits. This allows us to create a mapping
between an orbit at the start of the CE and the orbit at the end of the CE
phase. The formalism has the nickname SCATTER, for Single Components’ Angular
momenTum TransfER.

The SCATTER approach confers several advantages. First, it predicts changes in
the angular momentum of the stellar components and envelope that can be
compared with the results of simulations. This type of check can be used both
to understand simulation results and to refine the SCATTER formalism. Second,
the functional forms we derive are relatively simple, suggesting that they can
be successfully applied to a wide range of binary systems from those including
WDs to those including BHs. Finally, as mentioned above, the approach we
develop generalizes in a direct way to CEs in triples and higher-order
multiples. The reason for this is that SCATTER considers the angular momentum
transfer from each of the components of a multiple-star system. While the
formalism’s application to higher-order multiples requires that we develop
expressions for the amount of mass primarily exchanging angular momentum with
each component, SCATTER provides a robust framework for systems more complex
than we explicitly consider here.

We start in § 2 by discussing the motivation for a new formalism. In § 3 we
introduce our assumptions. We use them to derive a simple expression that maps
an initial state, i.e., at the time when the envelope is released by its star,
to a final state, i.e., when the binary has settled into its post-CE orbit.
This map involves free parameters. § 4 is devoted to the enterprise of
deriving physically and observationally reasonable parameter values, while in
§ 5 we find a formula that allows the key parameter to be selected in a manner
that is consistent with all the data we have and which is easy to apply. In §
6 we apply our formalism to randomly generated binaries, each of which
includes a WD. We compare the results with those derived using standard
implementations of the “$\alpha$” and “$\gamma$” formalisms in § 7. Section 8
is devoted to an overview of the work.

##  2 Motivation

Consider a binary whose components are close enough to each other that the
first-evolved star fills its RL after its core has a mass greater than about
$0.1\,\mathrm{M}_{\sun}$. If the mass transfer is unstable on a dynamical time
scale, the donor’s envelope becomes a CE, encompassing the core and its
companion star. The core and companion may merge. If, however, the orbit stops
shrinking, its orbital separation at the end of the CE phase, $a(f)$, is
generally small enough that the companion, Star 2, will eventually fill its
RL. Should Star 2 have a mass larger than that of the remnant of Star 1,
and/or be at least somewhat evolved, the second phase of mass transfer
initiated by the RL filling of Star 2 may also yield a CE. Many interacting
binaries thus experience two CE phases. Evolution with two CEs is particularly
common in binaries whose final fate111The merger may occur within the CE or it
may occur at a later time, after dissipative forces such as the emission of
gravitational radiation have had time to further shrink the orbit. is the
merger of two compact objects (e.g., Wu et al., 2018; Marchant et al., 2021).

The prevalence of CE evolution suggests that uncertainties in the CE phase
lead to uncertainties in the rates, times, and characteristics of many events
currently of great interest, such as the mergers of BHs, short gamma-ray
bursts, or the production of SNe Ia (e.g., Iben & Tutukov, 1984; Tutukov &
Yungelson, 1993). Such events are important: BH and/or NS mergers provide
direct insight into the properties of BHs and NSs. SNe Ia light up distant
parts of the universe, allowing us to study the accelerated expansion of the
Universe. SNe Ia also influence the host galaxies, for example the chemical
evolution of galaxies (GCE), which has been used for constraining the
formation and evolution of galaxies in the universe. Binary population
synthesis (BPS) is motivated by the goal of computing the number of events
(such as mergers, or SNe Ia), the time the events occur relative to star
formation, and the characteristics of the events (e.g., Belczynski et al.
2002; Ruiter et al. 2009; Mennekens et al. 2010; Mennekens & Vanbeveren 2014;
Kruckow et al. 2018; Han et al. 2020; Mandel et al. 2021; Briel et al. 2022;
see also Toonen et al. 2014; Mandel & Broekgaarden 2022 for a comparison). To
do this, a set of calculations starts with a population of binaries, and
evolves each to determine whether it becomes one of the systems of highest
interest. Because the CE is an evolutionary phase through which many of the
binaries pass, but which is notoriously difficult to constrain, it often
introduces a substantial uncertainty into BPS results.

Consider for example, the progenitors of SNe Ia. The nature or natures of SNe
Ia progenitors has been debated for more than half a century, but still there
is no perfect progenitor model. The two mainstream models can be described as
follows. In the double-degenerate (DD) model, two WDs in a close orbit
eventually merge and if the mass of the merger product reaches a critical
mass, often considered to be the Chandrasekhar mass (about
$1.4\,\mathrm{M}_{\sun}$) an SN Ia ensues (Neunteufel, 2020). In single
degenerate (SD) progenitor binaries, a WD has an extended companion whose
donated mass can bring the WD to the point of explosion. Both models are
incorporated into BPS simulations. See Jha et al. (2019); Soker (2019); Ruiter
(2020) for a review including the other scenarios (e.g., Raskin et al., 2009;
Rosswog et al., 2009; Ruiter et al., 2011; Foley et al., 2013; McCully et al.,
2014; Soker, 2015; Shen et al., 2018). Unfortunately, BPS calculations have so
far been unable to satisfactorily reproduce population properties such as
delay times and chemical enrichment of galaxies, see Figure 1 (see also
Kobayashi et al. 2022 for the details). Since $\alpha$ elements (e.g., O) are
mainly produced from core-collapse supernovae on a short timescale
($\mathrm{M}\mathrm{yrs}$) and the majority of Fe is produced from SNe Ia on a
longer timescale ($\gtrsim$0.1\text{\,}\mathrm{G}\mathrm{yrs}$$), the
[$\alpha$/Fe] ratio has been used as a proxy of ages of stellar populations in
galaxies. Although there are successful analytical models for including SNe Ia
in GCE (Greggio & Renzini, 1983; Matteucci & Greggio, 1986; Kobayashi et al.,
1998; Greggio, 2005; Kobayashi & Nomoto, 2009), there are no BPS models that
can reproduce the observations in the Milky Way galaxy (obtained from high-
resolution spectra of nearby stars Zhao et al., 2016), namely, the evolution
of elemental abundance ratios in the solar neighbourhood. The SN Ia delay
times are also important for constraining the star formation histories from
the observed [$\alpha$/Fe] ratios of early-type galaxies.

Figure 1: Comparison between observational data (black symbols) and specific
implementations of four BPS models. The red solid line uses the Brussels model
(Mennekens et al., 2010, 2012). The green long-dashed line uses StarTrack
(Ruiter et al., 2009, 2014). The blue short-dashed line uses ComBinE (private
communication based on Kruckow et al. 2018). The magenta dotted line uses
BPASS (Briel et al., 2022).

The uncertainties in the CE parameters are also responsible for significant
uncertainties in the rates of double BH mergers (e.g., Bethe & Brown, 1998;
Dominik et al., 2012; Chruslinska et al., 2018; Kruckow et al., 2018; Vigna-
Gómez et al., 2018; Mapelli & Giacobbo, 2018; Tanikawa et al., 2021; Bavera et
al., 2021; Olejak et al., 2021), for more details see e.g. Figures 3.47, B.48,
and B.49 in Kruckow (2018) and a recent summary in Mandel & Broekgaarden
(2022). This is because gravitational wave emission, hence the time between
star formation and merger, depend strongly on the orbital separation (Peters,
1964). Hence, the results of CE evolution have major impacts on the rates of
such merging events. For mergers involving NSs the uncertainties of CE
evolution are smaller but still important, while kicks during NS formation can
introduce significant eccentricity and also influence on the rate of mergers
that include NSs.

##  3 The Formalism

###  3.1 Basic assumptions

The SCATTER formalism aims to derive the final (post-CE) separation between
the binary components, and does not rely on understanding the mechanisms
ejecting the CE. The physics of CE evolution is complex. Both energy transfer
and angular momentum transfer almost certainly each take place through a
variety of mechanisms, and at a range of times from the start of the CE. For
example, angular momentum transfer to the envelope may occur at early stages,
before the companion enters the CE or when it is still in the outer envelope
(Soker, 2004). The SCATTER formalism makes no assumptions about the timing or
mechanisms responsible for the transfer of angular momentum, or about the
state of the CE and its reaction to interactions with the binary’s components.
We make only the following basic physical assumptions.

  * •

The initial state at the start of the CE is defined by the angular momentum of
the binary, which is the sum of the angular momenta of its components. The
same is true of the final state. Thus, the change between the initial and
final angular momentum of the binary’s components provides a map from the
binary’s initial state to its final state. We assume that the angular momentum
lost by each of the binary’s components during the CE was transferred from it
to the envelope. The effect of the stars on the envelope, and the mechanisms
leading to its ejection are not explicitly considered.

  * •

Angular momentum is conserved between the initial and final state of the
binary. That is, angular momentum lost by the binary is transmitted to the
envelope. The angular momentum transferred from each component of the binary
can be viewed as providing angular momentum to a quantitatively well-defined
fraction of the envelope. We do not attempt to identify the specific portions
of the envelope that receive angular momentum from each star. We do estimate
the quantity of mass that received angular momentum primarily from each
component star.

  * •

We take the initial state to be that in which the envelope has just been
released. The mass of the donor’s core and its companion are $M_{\mathrm{c}}$
and $M_{\mathrm{2}}$, respectively. The orbital radius is $a(0).$ The envelope
mass is $M_{\mathrm{e}}$. Star 1 is the donor, with pre-CE mass
$M_{1}=M_{\mathrm{e}}+M_{\mathrm{c}}$.

  * •

In this paper we do not consider mass exchange. In other words, the masses of
the two objects spiraling toward each other are constant. This restriction can
be lifted within the formalism by incorporating mass exchange in Equation (6).

  * •

The initial and final orbits of the binary are circular.

We also note that this formalism can be applied to take advantage of the fact
that the angular momentum is a three vector. In this paper, however, we
consider just a single component of the angular momentum sufficient to
describe a binary orbit. In addition, the formalism can be naturally extended
to predict end states of CEs that occur in higher order multiples. This is the
subject of a separate paper.

###  3.2 Derivation

The CE starts at the time the donor star, Star 1 with mass
$M_{\mathrm{1}}=M_{\mathrm{e}}+M_{\mathrm{c}}$, fills its RL. At the moment of
RL filling, the core mass of Star 1 is $M_{\mathrm{c}}$ and its envelope mass
is $M_{\mathrm{e}}$. The orbital separation between the stellar centers is
$a(0)$. On a dynamical time scale Star 1 loses its envelope. The stripped core
and Star 2 spiral toward each other within the envelope. Whatever the time
scale of the spiral-in or plunge-in, the CE itself takes some time to
dissipate. Ivanova et al. (2013) pointed out that roughly one in five
planetary nebulae may correspond to CEs. At typical expansion velocities in
the range of tens to a few hundred km s-1, it takes $10^{4}$ yr to more than
$10^{5}$ yr for the envelope to disperse. Our goal is to determine whether the
two masses merge while still inside the CE and, if they don’t, to find a
physically reasonable expression for their final orbital separation, $a(f)$.

####  3.2.1 $\Delta M_{\mathrm{e},i}$

All CE formalisms employ a fundamental principle to relate the final state to
the initial state. We use angular momentum conservation, assuming that the
evolution of the orbit can be computed by calculating the orbital angular
momentum each star imparts to some mass in the CE. Thus, Star $i$ exchanges
angular momentum with an amount of envelope mass $\Delta M_{\mathrm{e},i}$.
Note that $\Delta M_{\mathrm{e},i}$ refers to mass in the CE. The subscript
$i$ simply indicates that this amount of mass exchanged angular momentum
mainly with Star $i$. The following equation applies.

$\sum_{i}\Delta M_{\mathrm{e},i}=M_{\mathrm{e}},$ (1)

where the sum covers all of the stellar components. For binaries,
$i\in\\{\mathrm{c},2\\}$. This approach treats the CE as a union of distinct
sets, each of which draws angular momentum from a single star. This may
suggest that the envelope can be neatly divided into contiguous regions in
which one stellar component is primarily responsible for angular momentum
transmission. In fact, the situation can be highly complex, especially because
the positions and velocities of small masses in the envelope change in time.
Furthermore, bits of mass in the envelope interact with other bits of envelope
mass. In our formalism we assume that each small bit of envelope mass that has
gained angular momentum during the CE phase, drained angular momentum from the
binary’s components and that it is possible, perhaps only in a probabilistic
sense, to say that one of the binary’s component masses was the primary donor
of angular momentum to that bit of mass.

Figure 2: The influence of the choice of $\delta$. Top: $\mathcal{Q}$ for
different $\delta$s, which is colour coded (lines of even numbers of $\delta$
are solid and odd numbers are dashed). The part below the lines is the amount
of mass influenced by $M_{\mathrm{c}}$ (left y-axis) and above is the amount
of mass influenced by $M_{\mathrm{2}}$ (right y-axis). The plot is point
symmetric at $q=M_{\mathrm{c}}/M_{\mathrm{2}}=1$ and
$\mathcal{Q}(1,\delta)=0.5$. Middle: The angular momentum change caused by
$M_{\mathrm{c}}$ for the same functions as in the top panel. Bottom: The sum
of the angular momentum change by both stars,
$\mathcal{F}\left(M_{\mathrm{c}}/M_{\mathrm{2}},\delta\right)$. The gray area
marks $1.5\leq\delta\leq 3.5$.

We do not attempt to identify the source of angular momentum for each part of
the envelope or even to assign a probability of angular momentum transfer to
each of the binary components. Instead, we employ a simple expression that
emerges from the RL formalism to estimate the quantity, $\Delta
M_{\mathrm{e,c}}$ of the CE mass that drains orbital angular momentum
primarily from $M_{\mathrm{c}}$. Similarly, we estimate $\Delta
M_{\mathrm{e,2}}$, which drains orbital angular momentum primarily from
$M_{\mathrm{2}}$. Both mass and rotation play roles in determining the
effective potential. It is therefore useful to consider the RL of each star.
If $q\equiv M_{\mathrm{c}}/M_{\mathrm{2}}$, then the effective radius of the
core’s RL is $f(q)\,a$, where $a$ is the separation between Star 2 and Star c,
and

$f(q)=\frac{0.49\,q^{\frac{2}{3}}}{0.6\,q^{\frac{2}{3}}+\ln\left(1+q^{\frac{1}{3}}\right)},$
(2)

this function is called $r_{\mathrm{L}}$ in Eggleton (1983).

We define

$\mathcal{Q}\left(q,\delta\right)\equiv\frac{\left[f\left(q\right)\right]^{\delta}}{\left[f\left(q\right)\right]^{\delta}+\left[f\left(q^{-1}\right)\right]^{\delta}}.$
(3)

The parameter $\delta$ corresponds to a dimension, so that roughly speaking,
$[a\,f(q)]^{\delta}$ corresponds to a $\delta$-dimensional volume.

Note that $\mathcal{Q}(q,\delta)+\mathcal{Q}(q^{-1},\delta)=1$. We therefore
define

$\Delta M_{\mathrm{e,c}}\equiv\mathcal{Q}(q,\delta)\,M_{\mathrm{e}},$ (4)

to be the fraction of the CE mass draining orbital angular momentum primarily
from $M_{\mathrm{c}}$, with an analogous expression for $\Delta
M_{\mathrm{e,2}}=\mathcal{Q}(q^{-1},\delta)\,M_{\mathrm{e}}$.

The functional form of $\mathcal{Q}$ has the advantages of (1) assuring that
the more massive star interacts with a larger portion of the CE; (2) having a
symmetric form; and (3) being normalized. The applicability of this function
can be tested through comparisons with observations. Future studies may alter
the definition of $\mathcal{Q}$, perhaps to better represent observational
constraints.

####  3.2.2 Angular Momentum

The orbital angular momentum may be written as

$L=M_{\mathrm{1}}\,M_{\mathrm{2}}\,\sqrt{\frac{G\,a}{M_{\mathrm{b}}}},$ (5)

where $M_{\mathrm{b}}$ refers to the total mass of the binary at the start of
the CE: $M_{\mathrm{b}}\equiv M_{\mathrm{1}}+M_{\mathrm{2}}$, where
$M_{\mathrm{1}}=M_{\mathrm{c}}+M_{\mathrm{e}}$, $G$ is the gravitational
constant, and $a$ is the current semi-major axis of a circular orbit. As the
CE starts, $a=a(0)$. During the CE phase, the value of the $M_{\mathrm{c}}$
stays the same, but the envelope is lost, effectively decreasing the total
mass $M_{\mathrm{b}}$ of the binary as well as the mass of Star 1. The orbital
separation also shrinks. The expression for orbital angular momentum, $L$,
leads to the following equation.

$\frac{\mathrm{d}L}{L}=\frac{1}{2}\,\frac{\mathrm{d}a}{a}-\frac{1}{2}\,\frac{\mathrm{d}M_{\mathrm{b}}}{M_{\mathrm{b}}}+\frac{\mathrm{d}M_{\mathrm{1}}}{M_{\mathrm{1}}}+\frac{\mathrm{d}M_{\mathrm{2}}}{M_{\mathrm{2}}}.$
(6)

This expression is completely general. If the three-dimensionality of the
angular momentum is important, we have three such equations, one for each
spatial direction. Here we make the simplification of treating the angular
momentum as a scalar quantity, which is equivalent to assuming the the vector
normal to the orbital plane points in the same direction at all times.

If known external torques act on the binary, we can compute the quantity
$dL/L$. To find an expression for $dL/L$ we note that the CE exerts a torque
on each component of the binary. Each binary component mass therefore
experiences a change in its angular momentum. For the core, we have

$\frac{dL_{\mathrm{c}}}{L}\,=\eta_{\mathrm{c}}\,Q\left(\frac{M_{\mathrm{c}}}{M_{\mathrm{2}}},\delta\right)\,\frac{dM_{\mathrm{e}}}{M_{\mathrm{c}}}\,\frac{M_{\mathrm{2}}}{M_{\mathrm{c}}+M_{\mathrm{2}}}.$
(7)

This equation defines a quantity $\eta_{\mathrm{c}}$: the fractional change in
$L_{\mathrm{c}}$ is equal to the specific angular momentum of the core times
the amount of envelope mass with which it interacts, times the constant of
proportionality $\eta_{\mathrm{c}}$.

For Star 2 we have

$\frac{dL_{\mathrm{2}}}{L}\,=\eta_{\mathrm{2}}\,Q\left(\frac{M_{\mathrm{2}}}{M_{\mathrm{c}}},\delta\right)\,\frac{dM_{\mathrm{e}}}{M_{\mathrm{2}}}\,\frac{M_{\mathrm{c}}}{M_{\mathrm{c}}+M_{\mathrm{2}}}.$
(8)

Thus, the quantity on the left-hand side of equation (6) can be expressed as
follows:

$\frac{\mathrm{d}L}{L}=\frac{\mathrm{d}L_{c}}{L}+\frac{\mathrm{d}L_{2}}{L}.$
(9)

With an explicit expression for $dL/L$, we can plug into equation (6) derive
the functional form of $da/a$.

$\frac{\mathrm{d}a}{a}=2\,\Bigg{[}\frac{\mathrm{d}L_{c}}{L}+\frac{\mathrm{d}L_{2}}{L}\Bigg{]}+\frac{dM_{\mathrm{b}}}{M_{\mathrm{b}}}-2\,\frac{dM_{1}}{M_{1}}-2\,\frac{dM_{2}}{M_{2}}.$
(10)

Integration from the initial to final masses then yields $a(f)/a(0)$ or its
inverse. In this integration we assume that Star 2 does not gain mass during
the CE. This assumption can easily be relaxed be relaxed. When, however, there
is no significant accretion onto the companion, then
$\int\mathrm{d}M_{\mathrm{2}}\approx 0$. Similarly, $M_{\mathrm{c}}$ remains
constant. The binary loses the mass $M_{\mathrm{e}}$, the same as the mass
lost by Star 1, thus $\int\mathrm{d}M_{\mathrm{1}}\approx-
M_{\mathrm{e}}\approx\int\mathrm{d}M_{\mathrm{b}}$.

This procedure yields

$\displaystyle\left(\frac{a(0)}{a(f)}\right)\,\left(1+\frac{M_{\mathrm{e}}}{M_{\mathrm{c}}}\right)^{2}\,\left(1+\frac{M_{\mathrm{e}}}{M_{\mathrm{c}}+M_{\mathrm{2}}}\right)^{-1}=$
(11)
$\displaystyle\exp\left(\left[\frac{2\,M_{\mathrm{e}}}{M_{\mathrm{c}}+M_{\mathrm{2}}}\right]\,\left[\eta_{\mathrm{c}}\,\frac{M_{\mathrm{2}}}{M_{\mathrm{c}}}\,Q\left(\frac{M_{\mathrm{c}}}{M_{\mathrm{2}}},\delta\right)+\eta_{\mathrm{2}}\,\frac{M_{\mathrm{c}}}{M_{\mathrm{2}}}\,Q\left(\frac{M_{\mathrm{2}}}{M_{\mathrm{c}}},\delta\right)\right]\right).$

The ratio of the initial to final orbital separations depends only on mass
ratios: $q=M_{\mathrm{c}}/M_{\mathrm{2}}$, and its inverse, and
$q_{\mathrm{ec}}\equiv M_{\mathrm{e}}/M_{\mathrm{c}}$. Note that the
$M_{\mathrm{e}}/(M_{\mathrm{c}}+M_{\mathrm{2}})$ can be written as

$\frac{M_{\mathrm{e}}}{M_{\mathrm{c}}+M_{\mathrm{2}}}=q_{\mathrm{ec}}\,\frac{q}{1+q}.$
(12)

Dependence on only mass ratios suggests applicability to a wide range of
systems across mass ranges. The specific value of the donor’s starting stellar
mass comes into play only through the value of $a(0)$, since

$a(0)=\frac{R_{\mathrm{1}}(0)}{f\left(\frac{M_{\mathrm{1}}}{M_{\mathrm{2}}}\right)},$
(13)

where $R_{\mathrm{1}}(0)$ and $M_{\mathrm{1}}$ are, respectively, the radius
and mass of the donor star at the time of RL filling.

A convenient way to write Equation (11) is as follows:

$\displaystyle a(f)=$ (14) $\displaystyle
a(0)\,\left[\frac{(1+q)\,(1+q_{\mathrm{ec}})^{2}}{1+q\,(1+q_{\mathrm{ec}})}\right]\,\exp\left[-2\,\eta\,\left(\frac{q\,q_{\mathrm{ec}}}{1+q}\right)\,\mathcal{F}(q,\delta)\right],$

where

$\mathcal{F}\left(q,\delta\right)=\mathcal{Q}\left(q,\delta\right)\,\frac{1}{q}+\mathcal{Q}\left(\frac{1}{q},\delta\right)\,q.$
(15)

Here we simplified the equation (15) by assuming that the proportionality
parameter is the same for both stellar components,
$\eta=\eta_{\mathrm{c}}=\eta_{\mathrm{2}}$.

By itself, $Q(q,\delta)$ is a measure of only how the mass ratio influences
the amount of envelope mass accepting orbital angular momentum from each
component of the binary. The function $\mathcal{F}$ is a sum of terms, one of
which is illustrated in the middle panel. The combined effects of the two
terms forming $\mathcal{F}$ has a kind of self regulation effect, as shown in
the bottom panel of Figure 2. Nevertheless, across values of the mass ratio,
$\mathcal{F}$ can achieve a wide range of values depending on $\delta$.
Extreme values of $\delta$ could therefore produce extreme values of $a(f),$
unless we were to invoke extreme values of $\eta.$ On the other hand, for
values of $\delta$ in the vicinity of $2$, there is little variation on the
value of $\mathcal{F}$ as $q$ varies over almost $4$ orders of magnitude.
Whatever the value of $\delta$, there is an interval around $q=1$ for which
the value of $\mathcal{F}$ is unity; the size of this region is also smaller
for large values of $\delta$.

###  3.3 Generalizations

The derivation above can be extended in several ways. One way would be to make
different choices for the functions that describe quantities such as the
amount of CE mass to which each binary component transfers angular momentum.
Another function could, for example, be used in place of the function
$Q(q,\delta)$. Simulations of CEs may suggest alternatives. If the functional
form changes, the parameters employed in the function would likely be differed
as well. Thus, the basic ideas of SCATTER as outlined in §3.1 can be
implemented in different ways. We note that the choices we have made are
physically motivated. Furthermore, we show in §4 that they provide a
consistent set of predictions for large sets of known post CE binaries.

The second type of extension can be carried out using our functional choices,
or using others. These include, for example, (1) allowing the components to
gain or lose mass during the CE; (2) treating the angular momentum as a
vector, which would have the physical effect of introducing spin to the post
CE binaries; (3) considering systems with higher multiplicity; (4) using post-
CE binaries of types different from those we employ in the later sections of
the paper. With regard to the latter point, such systems could include
planetary systems that have experienced CEs, and also systems in which the
compact objects are BHs and NSs, without a WD component.

##  4 The Parameters

Our implementation of the SCATTER formalism includes three parameters:
$\delta$, $\eta_{\mathrm{c}}$, and $\eta_{\mathrm{2}}$. Given values of these
parameters, we could start with any initial CE state and derive the final
orbital separation. In this section we discuss the roles of the parameters
within the formalism and try to gain insight from observations of CE final
states in order to estimate values of those parameters.

For now we consider that both binary components transmit a similar proportion
of their specific angular momentum to the CE. Thus:
$\eta=\eta_{\mathrm{c}}=\eta_{\mathrm{2}}$. This leave $\eta$ and $\delta$ as
the to-be-determined quantities. The parameter $\eta$ has a physical meaning,
because $\eta$ is the parameter which multiples the specific angular momentum
imparted to mass in the envelope. The components of the binary impart a larger
portion of their specific angular momentum to the envelope for larger values
of $\eta$. We may think of it as a sort of inverse efficiency parameter. Given
this, both very large and very small values of $\eta$ should be excluded. We
might expect that roughly $0.1<\eta<10$. Even employing the simple assumption
that $\eta$ is the same for both binary components, it is natural to
anticipate that its value depends on physical characteristics of the
components, in particular their mass ratio. The value of $\eta$ is also
expected to depend on the properties of the envelope, in particular the
envelope mass relative to the masses of the components. The form of Equation
(14) then tells us that, if $\eta=\eta(q,q_{\mathrm{ec}})$, then the value of
the final orbital separation depends only on $q$ and $q_{\mathrm{ec}}$. In the
absence of magnetic fields or external torques, the spin-down of the binary is
governed by just these two variables.

The value of $\delta$ determines how much of the envelope’s mass draws angular
momentum from each component of the binary. It may be thought of as a
dimensionality. Equation (3) shows that the larger the value of $\delta$, the
larger the value of $Q$ associated with the most massive binary component
(i.e., $f(q)$ increases with the mass ratio, $q$). The dependence of $Q$ on
$q$ and $\delta$ is illustrated in the top panel of Figure 2. We have
considered values of $\delta$ extending from $-5$ to $5$. The rough
correspondence with a dimensionality suggests that the value of $\delta$
should be positive. Although there is not an exact correspondence between the
amount of mass influenced and a volume whose size is determined by $\delta$,
we might expect values of $\delta$ to be near $3$ and will be discussed in
more detail in § 4.3.

###  4.1 Post-CE Binaries

Table 1: An overview of the systems selected from Kruckow et al. (2021). For each type of binary the number of the systems and the ranges for the main parameters (period and masses of the two stars) are quoted. Type | # | Period | Mass of WD | Mass of Comp.
---|---|---|---|---
|  | $[\mathrm{days}]$ | $[\mathrm{M}_{\sun}]$ | $[\mathrm{M}_{\sun}]$
WD+MS | $\\!187\\!$ | $\\!0.0651-88.1805\\!$ | $\\!0.07-1.23\phantom{0}\\!$ | $\\!0.087-3.4\phantom{0}\\!$
WD+NS | $\\!\phantom{0}28\\!$ | $\\!0.1024-95.1741\\!$ | $\\!0.16-1.37\phantom{0}\\!$ | $\\!1.14\phantom{0}-3.4\phantom{0}\\!$
WD+WD | $\\!\phantom{0}34\\!$ | $\\!0.0048-\phantom{0}2.21\phantom{00}\\!$ | $\\!0.15-0.813\\!$ | $\\!0.167-1.06\\!$

We use the database of hundreds of post-CE binary candidates compiled in
Kruckow et al. (2021). This catalog summarizes several decades of literature
about the observation of short period binaries which are potential end states
of CE evolution. The most useful information, including mass estimates,
originates from observations of eclipsing binaries and/or spectroscopic
observations. In Kruckow et al. (2021) the candidates for post-CE binaries are
selected by their present-day short orbital period (below
$100\text{\,}\mathrm{days}$). Here we require that at least one component be a
WD, and that there are mass estimates for each of the post-CE binary’s
components. Hence we exclude all flagged systems222Most flags indicate that
some of the data is assumed instead of measured or the binary has evolved
significantly since the ejection of the CE. and systems with only lower or
upper mass limits. In Table 1 we show the properties of the post-CE binaries
we utilize in this work.

###  4.2 Input from Stellar and Binary Evolution

To use the post-CE binaries to create a map between CE initial and final
states, we invoke results from both stellar and binary evolution. Stellar
evolution relates the donor’s radius to its evolutionary state. We consider
only donors whose cores will become WDs. In these cases, the simplest
prescription for the donor’s radius, $R_{1}(0)$ depends primarily on
$M_{\mathrm{c}}$ (Joss et al., 1987). At $t(0)$, the moment of RL filling,
when the core mass is $M_{\mathrm{c}}$, $M_{\mathrm{1}}$ may be close in value
to the donor’s zero-age main sequence (ZAMS) mass if there have not yet been
significant winds, but it may also be significantly smaller. In either case,
the mass of the envelope is $M_{\mathrm{e}}=M_{\mathrm{1}}-M_{\mathrm{c}}$.
Furthermore, when the core mass is larger than roughly $0.2\,M_{\odot}$, the
stellar radius has only a weak dependence on the initial mass of the donor. We
therefore select possible donor initial masses to be approximately equal to
$M_{\mathrm{e}}$. While detailed evolutionary calculations would refine these
estimates, a simple unified approach allows us to focus on the effects of CE
evolution.

At the time of Roche-lobe filling, the orbit is circular, with radius given by
the equation (13). Thus, if, in addition to having post-CE masses, we also
knew the value of $M_{\mathrm{e}}$, we could use the equation (14) to compute
$a(f).$

To select a possible value (or values) of $M_{\mathrm{e}}$, we used stellar
models to compute both its minimum and maximum possible values (Appendix A).
We estimated the minimum envelope mass to be
$(0.9\,\mathrm{M}_{\sun}-0.85\,M_{\mathrm{c}})$. If the result was an envelope
less massive than $0.6\,\mathrm{M}_{\sun}$, we instead set
$M_{\mathrm{e}}^{\mathrm{min}}$ to $0.6\,\mathrm{M}_{\sun}$. For WDs with mass
greater than $0.55\,\mathrm{M}_{\sun},$ we took the
$M_{\mathrm{e}}^{\mathrm{min}}$ to be
$(0.8\,\mathrm{M}_{\sun}+2.1\,M_{\mathrm{c}})$. For all WD’s we took the
maximum envelop mass, $M_{\mathrm{e}}^{\mathrm{max}}$ to be
$(1.3\,\mathrm{M}_{\sun}+10.5\,M_{\mathrm{c}})$. If, however,
$M_{\mathrm{e}}^{\mathrm{max}}$ was larger than $8\,\mathrm{M}_{\sun}$, we
instead set it to $8\,\mathrm{M}_{\sun}$.

The effective radius of the donor’s RL at the start of the CE is
$R_{\mathrm{RL}}(0)=a(0)\times
f(\frac{M_{\mathrm{e}}+M_{\mathrm{c}}}{M_{\mathrm{2}}})$. We can also define a
“final” effective RL radius to be $R_{\mathrm{RL}}(f)=a(f)\times
f(\frac{M_{\mathrm{c}}}{M_{\mathrm{2}}})$. This allows us to introduce a
variable

${\cal
R}_{\mathrm{d}}\equiv\frac{R_{\mathrm{RL}}(0)}{R_{\mathrm{RL}}(f)}=\frac{a(0)}{a(f)}\times\frac{f(\frac{M_{\mathrm{e}}+M_{\mathrm{c}}}{M_{\mathrm{2}}})}{f(\frac{M_{\mathrm{c}}}{M_{\mathrm{2}}})}.$
(16)

Large values of ${\cal R}_{\mathrm{d}}$ correspond to cases in which the CE
phase significantly shrunk the binary’s orbit. When we produce graphs of post-
CE binaries, we mark in green those with ${\cal R}_{\mathrm{d}}>100;$ these
are almost certainly CE end states. Binaries with $100>{\cal
R}_{\mathrm{d}}>20$ are probable CE end states, and are shown as blue points.
This with ${\cal R}_{\mathrm{d}}<20$ are red, and may not represent true CE
end states.

At time $t(f),$ when the CE phase ends, the orbital separation is $a(f)$. The
orbit generally continues to evolve. If, for example, the post-CE binary
comprised of compact objects and if it is small enough that gravitational
radiation carries away significant angular momentum, the binary continues to
shrink. The time to merger can then be estimated to be333We assume that $a(f)$
is much larger than the separation at which one of the post-CE binary’s
components will fill its RL.

$\tau_{\mathrm{merge}}=$1.5\times
10^{8}\text{\,}\mathrm{yrs}$\left[\frac{a(f)}{\mathrm{R}_{\sun}}\right]^{4}\left[\frac{\mathrm{M}_{\sun}^{3}}{M_{\mathrm{c}}\,M_{\mathrm{2}}\,M_{\mathrm{t}}}\right],$
(17)

where the total mass of the binary system at the end of CE is defined as
$M_{\mathrm{t}}\equiv M_{\mathrm{c}}+M_{\mathrm{2}}$. If the orbit is small
but the WD’s companion is a main sequence (MS) star, magnetic braking (rather
than gravitational radiation) may bring the binary’s components closer
together, and an epoch of mass transfer can ensue. If the mass transfer is
stable, the system will eventually become a cataclysmic variable, in which a
WD receives mass at a low rate from a close companion. In this case, the MS
star fills its RL. In the figure of Section 4.4.1, if the value of $a(f)$ in
the post-CE state is such that the MS star is close to Roche-lobe filling, we
mark the system with a special symbol, a large gold pentagon, to indicate that
the post-CE binary may have evolved, so that the binary separation we measure
is not the same as the CE end state.

###  4.3 Constraining $\delta$

Our first step is to estimate the value of $\delta$, which appears only in the
factor $\mathcal{F}(q,\delta)$. The bottom panel of Figure 2 plots the
function $\mathcal{F}(q,\delta)$ as a function of $q$ for the full range of
$q$ we need to consider. Each curve employs a single value of $\delta$,
ranging from -5 [dashed, blue top-most curve] to 5 [dashed red bottom-most
curve]. Note that, for both the smallest and largest values of $\delta$, the
value of $\mathcal{F}(q,\delta)$ varies over more than an order of magnitude.
This suggests that $\eta$ would also be highly variable if $\delta$ is either
very large or small. Since $\delta$ defines the dimensionality of the volume
of matter interacting with the binary components, it seems likely that its
value should lie between 1 and 4. The gray area in the bottom panel of Figure
2 shows the region in which $\delta$ ranges between 1.5 and 3.5. The solid
(dashed) line in that region corresponds to $\delta=2$ ($\delta=3$). For
values of $\delta$ in this more limited range, the variation in $\eta$ from
binary to binary would not have to be large because the variation of
$\mathcal{F}(q,\delta)$ is modest over the range of likely $q$ values.

Figure 3: Values of $\log_{10}[\eta]$ are plotted as functions of $\delta$.
Eighteen different values of $\delta$ were considered from $0.5$ to $9.0$. For
each value of $\delta$, we consider four different values of $M_{\mathrm{e}}$,
the envelope mass. Each value of $M_{\mathrm{e}}$ is represented by 2 curves
of a distinct color. For each pair, we computed $\eta$ for every WD-MS post-CE
binary, and recorded the maximum and minimum values, $\eta_{\mathrm{max}}$ and
$\eta_{\mathrm{min}}$, respectively. Plotted are $\eta_{\mathrm{max}}$ and
$\eta_{\mathrm{min}}$ for each value of $\delta$.

To explore the influence of $\delta$, we consider post-CE binaries containing
a MS star in close orbit with a WD. In this case, our initial list, drawn from
the post-CE catalog, consisted of 112 binaries. First we hold the envelope
mass constant and explore the variation in $\eta$ among the post-CE binaries.
For the purpose of computing $R_{1}(0)$, which is only weakly dependent on the
initial donor mass, we use $R_{1}(0)(M_{\mathrm{ZAMS}},M_{\mathrm{c}})$ from
Joss et al. (1987), where we set $M_{\mathrm{ZAMS}}$ to the sum of the core
mass and the envelope mass. While the true value of $M_{\mathrm{ZAMS}}$ could
be larger if winds were active prior to RL filling, the trends we identify are
not altered for different choices. In this set of calculations we vary the
envelope mass between $1\,\mathrm{M}_{\sun}$ and $4\,\mathrm{M}_{\sun}$. We
varied the value of $\delta$ from $0.5$ to $9$ in increments of $0.5$. For
each value of $\delta$ we computed $\eta$ for each post-CE binary. We kept
track of the minimum value of $\eta$, $\eta_{\mathrm{min}}$ and its maximum
value $\eta_{\mathrm{max}}$. The results are shown in Figure 3. Note that for
larger values of $\delta$, the factor
$\mathcal{F}(\frac{M_{\mathrm{c}}}{M_{\mathrm{2}}},\delta)$ becomes smaller,
so that the value of $\eta$ must be larger. Although we do not know exactly
what values of $\eta$ should be assumed, we have noted that we expect it to be
larger than roughly $0.1$ and smaller than $10$. Figure 3 indicates that
$\eta_{\mathrm{max}}$ begins to undergo a sharp increase at values of $\delta$
larger than $3$. The values of $\eta_{\mathrm{min}}$ decrease for lower values
of $\delta$. Thus, suggests that we should use $\delta\approx 3$. We will we
employ $\delta=3$ throughout the remainder of the calculations described in
this paper. We note, however, that the choice is not unique.

###  4.4 The Value of $\eta$

The value of $\eta$ depends on the mass ratios,
$q=M_{\mathrm{c}}/M_{\mathrm{2}}$ and
$q_{\mathrm{ec}}=M_{\mathrm{e}}/M_{\mathrm{c}}$. Because the exponent contains
the factor $\eta\times\frac{M_{\mathrm{e}}}{M_{\mathrm{t}}}=\eta\times
q_{\mathrm{ec}}q/(1+q)$, where $M_{\mathrm{t}}=M_{\mathrm{c}}+M_{\mathrm{2}}$,
we can see that, for a given post-CE binary with an unknown value of
$M_{\mathrm{e}}$, larger values of $M_{\mathrm{e}}$ may be associated with
smaller values of $\eta.$ This is simply a statement that, to produce a given
final state, the specific angular momentum imparted to a portion of the
envelop can be smaller when the total envelop mass is larger. The dependence
of the factors outside the exponent means that the relationship between $\eta$
and $M_{\mathrm{e}}$ is somewhat more complicated, as we discuss in the
context of different types of post-CE binaries.

We use observed post-CE binaries to estimate values of $\eta.$ The largest
group of post-CE binaries for which we have reliable information is the set of
187 WD-MS binaries. In §4.4.1 we focus on them, and show similar figures for
WD-WD and WD-NS systems in Appendix B.

####  4.4.1 White-Dwarf/Main-Sequence Binaries

Figure 4: Relationships between the parameters of WD+MS post-CE binaries in
the observational data catalogue from Kruckow et al. (2021). Green points
correspond to binaries that have experienced significant shrinkage during a
prior CE phase. Blue points have experienced more modest shrinkage, and red
points have had little or no shrinkage. The quantities plotted are the WD and
MS masses (in solar masses), and the logarithm to the base 10 of the final
orbital separation (in solar radii). Also shown are
$\log_{10}[\mathcal{R}_{\mathrm{d}}]$ (see equation (16)), a measure of
shrinkage during the CE, and $\log_{10}[\mathcal{R}_{\mathrm{MS}}]$ (see
equation (20)), a measure of whether the binary is close to filling its RL
today. The values of $\eta$ have been estimated. Please see the text for more
detailed explanations of the variables.

We started with 187 WD-MS-star post-CE binaries (see Table 1). To gain some
insight into how these binaries were affected by the CE, we considered their
possible initial states, i.e., the state before the binary shrank. For
binaries in which a WD is the remnant of the RL filling companion, the initial
radius of the donor, $R_{1}(0)$ must have been larger than its MS radius.

The binary’s orbital separation at that time was
$a(0)=R_{1}(0)/f(M_{\mathrm{1}}/M_{\mathrm{2}})$. We assume that the value of
$M_{\mathrm{2}}$ does not change during the CE. $M_{\mathrm{1}}$, the mass of
the donor, is the sum of the core mass, which is the mass of today’s WD, and
the envelope mass.

By estimating the value of $M_{\mathrm{e}}$, we can estimate the value of
$a(0)$. To do this, we have simply used the mean value of the possible range
of envelope masses

$M_{\mathrm{e}}=0.5\,\left(M_{\mathrm{e}}^{\mathrm{min}}+M_{\mathrm{e}}^{\mathrm{max}}\right),$
(18)

where $M_{\mathrm{e}}^{\mathrm{min}}$ and $M_{\mathrm{e}}^{\mathrm{max}}$ are
defined at the conclusion of § 4.2. Although these are not likely to be the
true values of the envelope masses, by making the choice in the same way for
all post-CE binaries we can make comparisons and examine trends. In Figure 4,
the vertical axis in the top row of panels plots a variable
$\mathcal{R}_{\mathrm{d}}$. Note that some of the red points appear to
correspond to cases in which the radius increased. This could be due to
uncertainties in the measured system parameters or in the applicability of our
assumptions. Nevertheless, we cannot use the systems shown in red as good
examples of post-CE binaries. Systems shown in blue have values of the ratio
between 20 and 100. Some, possibly all of these, may have passed through a CE
phase. Points in green have experienced shrinkage by a factor of 100 or more
and are therefore the strongest candidates for post-CE systems.

In addition to these three colors we see that some points are surrounded by a
golden pentagon. In these systems, the present-day separation is close enough
that the MS star is filling or nearly filling its RL. While it is possible
that the CE ended at the stage in which mass transfer from the MS star could
start, this would appear to be a coincidence. More likely, the orbital
separation shrank during a possibly long time interval after the CE had
ceased. Thus, the systems represented by points surrounded by golden pentagons
may not represent true post-CE states, but may instead be states that evolved
after the end of the CE. Let $R_{\mathrm{RL}}^{\mathrm{MS}}$ represent the RL
radius of the companion star with $M_{\mathrm{2}}$,

$R_{\mathrm{RL}}^{\mathrm{MS}}=\frac{a(f)}{f(M_{\mathrm{2}}/M_{\mathrm{c}})}.$
(19)

We use $M_{\mathrm{MS}}=M_{\mathrm{2}}$, the value of the MS-star’s mass, as a
proxy for the star’s radius. Small values of the ratio

$\mathcal{R}_{\mathrm{MS}}\equiv\left(\frac{R_{\mathrm{RL}}^{\mathrm{MS}}}{M_{\mathrm{MS}}}\right){\rm~{}~{}~{}\left[\frac{R_{\odot}}{M_{\odot}}\right]}$
(20)

are not favored for pure post-CE states. Thus, while we explore the possible
evolution of all systems in the sample given in Table 1, we view as pure CE
survivors only the those green (and possibly blue) points not enclosed by
pentagons, which correspond to systems having $\mathcal{R}_{\mathrm{MS}}>1.5$.

We examine the trends exhibited by the distribution of systems in the panels
of Figure 4, and what they tell us about the past and future evolution of
these post-CE binaries. The vertical axis of the top set of panels shows
$\mathcal{R}_{\mathrm{d}}$, the fractional shrinkage during the CE phase.
Because of their definitions, the red, blue, and green points are in the
bottom, middle, and top regions of the plot, respectively, corresponding to
systems exhibiting the least (bottom) to most (top) shrinkage. The horizontal
axis of the top left panel shows the ratio $\mathcal{R}_{\mathrm{MS}}$.
Although there is a clear vertical grouping of systems that may be undergoing
(or be close to undergoing) mass transfer, there are also systems (for each of
the three colors) that are not currently interacting.
$\mathcal{R}_{\mathrm{MS}}$ is also plotted along the vertical axis of the
second row of panels. The left-most panel in the second row shows that there
is a clear separation between binaries with MS stars having masses above
$1\,\mathrm{M}_{\sun}$ and those with masses below roughly
$0.6\,\mathrm{M}_{\sun}$. The green systems tend to house those MS stars of
lower mass. In the middle panel of the second row we see that there is also a
distinction between systems with higher-mass and lower-mass WDs, although
these groups are not separated by a gap. Clear CE survivors (points in green)
tend to have higher WD mass; note that some of this effect may be associated
with their initially wider orbits.

This distinction is even clearer in the third-row plot of $M_{\mathrm{MS}}$
versus $M_{\mathrm{WD}}$. The clear CE survivors (green) tend to have higher-
mass WDs and lower-mass companions. Their values of $q$ therefore are
generally larger than for the systems that have experienced less shrinkage.
Note however, that although the existence of clusters of points separated by
mass is distinctive, it is not universal. There are some points spanning the
gap. By studying this panel, and the panels in which the component masses are
plotted against the final (present-day) orbital separation, we can see that
the systems lying between clusters of points may have important roles in the
future evolution of binary populations. This is because they have WDs massive
enough to be pushed to the Chandrasekhar mass by their companions, which could
potentially each shed several tenths of a solar mass. We cannot say for
certain that any of the WDs in this sparse set of systems will undergo
accretion-induced collapse or SNe Ia, but they are candidates for binaries
that can do so via accretion. We must therefore ensure that any CE formalism
is capable of predicting these systems and their numbers relative to other WD-
containing binaries.

Analyzing the binaries in the clusters that appear in these plots, we see that
systems in red tend to have stars between $0.5\,\mathrm{M}_{\sun}$ and
$3.5\,\mathrm{M}_{\sun}$ in orbit with helium WDs. Some of the orbits are
close enough that mass transfer can start soon if it has not already started.
Because the mass of the future donor (today’s MS star) is generally much
higher than the mass of the helium WD, we expect that when the donor fills its
RL, there will be a CE. Depending on the state of the donor star when that
occurs, the result could be a system with either two helium WDs or a carbon
oxygen and helium WD, which eventually merge subsequently (Dan et al., 2014).
A similar fate could await some of the red-point systems with larger present
day orbital separations. Systems in green have high mass WDs but their
companion MS stars are of such low mass that they do not have much mass to
donate. We would not expect the stars to experience a second CE phase, because
the potential donor star is less massive than the mass of the potential
accretor. These stars will likely become cataclysmic variables. Some of them
may already be cataclysmic variables.

We can also study the post-CE groups with the goal of learning about the CE
phase that produced them. The systems in red were produced when a star with a
helium core filled its RL. The shrinkage may have been minimal because the
mass of the donor star was less than that of the companion, today’s MS star.
Or it could have been the case that the small mass ratio,
$M_{\mathrm{1}}/M_{\mathrm{2}}$, was associated with conditions that drive out
the CE efficiently. Or else, the relatively dense envelope associated with a
low-mass core in a donor of modest mass may be associated with conditions that
are more easily driven out. The points in green, which experienced the most
shrinkage had more evolved cores. The giant donor therefore evolved from stars
that were initially more massive and were likely to be larger and less dense.
The mass ratio was also larger.

####  4.4.2 Possible Values of $\eta$

The fourth column of Figure 4 plots as a function of $\eta,$ each of the
quantities considered along the vertical axes described above
$(\mathcal{R}_{\mathrm{d}},\mathcal{R}_{\mathrm{MS}},M_{\mathrm{MS}},M_{\mathrm{WD}})$.
To compute the value of $\eta$ for an individual system, we needed to select a
value for $M_{\mathrm{e}}$. In these panels we used
$M_{\mathrm{e}}=0.5\,(M_{\mathrm{e}}^{\mathrm{min}}+M_{\mathrm{e}}^{\mathrm{max}})$.
In each panel, we see that the separation between the clusters in other panels
is exhibited here as well. The certain CE survivors (green) have usually
smaller values of $\eta$ than the systems in red that appear to have shrunk by
smaller factors. Most of the points in green are smoothly spread along a curve
whose smallest $\eta$ value is approximately $0.5$ and whose largest $\eta$
value is about $3$. Points in red have higher values, ranging up to about $7$.

We have carried out a parallel analysis of post-CE binaries that consist of
NS-WD pairs, and also post-CE end states that consist of WD-WD pairs. The
graphs are shown in the Appendix B. Because, as Table 1 shows, the numbers of
systems in these categories are relatively small, detailed conclusions cannot
be drawn. We note, however, that the values of $\eta$ are similar.

####  4.4.3 Constant Values of $\eta$

If we wanted our formalism to apply to only the clearest examples of CE end
states that form a relatively isolated clump, we could choose values of $\eta$
between $0.5$ and $2$. Under the assumption that trends discerned by using our
chosen envelope masses are robust, this would leave out the CE end states
that, as described above, may be related to accretion-driven SNe Ia or
accretion-induced collapse. It would also leave out those binaries which may
be genuine CE end states, but which did not shrink as much. Indeed, the trends
are likely to be robust, because, while different choices of envelope mass
would move some of the points, the effect would be modest, since a factor of
$n$ change in $\eta$ generally requires a multiplicative change of similar
magnitude in $M_{\mathrm{e}}$.

Figure 5: Curves in red correspond to systems that merge within $2\times
10^{5}\text{\,}\mathrm{yrs}$ in our simulation; systems represented in green
which merge within a Hubble time; in black are systems that require times
longer than the Hubble time to merge. The value of $M_{\mathrm{2}}$ is set to
be $0.8\text{\,}\mathrm{M}_{\sun}$. The value of $\eta$ was held fixed for a
simulation of $10^{6}$ CE evolutions, as described in the text. Figure 6:
Similar to Figure 5 but with $M_{\mathrm{2}}=$5\text{\,}\mathrm{M}_{\sun}$$.

We have therefore conducted a set of simulations that shows how different
values of $\eta$ influence CE end states. Starting with a fixed value of the
companion’s mass, $M_{\mathrm{2}}$, we consider a population of $\sim 10^{6}$
binaries that will undergo a CE phase. We take $\eta$ to have a constant value
and conduct the same simulation for 33 different values of $\eta$. This allows
us to assess the effects of selecting different values of $\eta.$ In order to
quantify the result, we focus on estimating the rates of gravitational mergers
post CE. At the time the envelope is released, the mass of the donor star,
$M_{\mathrm{1}}$, is the sum $M_{\mathrm{c}}+M_{\mathrm{e}}$. The companion
has mass $M_{\mathrm{2}}$, which is assumed to remain constant during the CE.
We begin the calculation by selecting a specific value for the companion’s
mass $M_{\mathrm{2}}$ to be $0.8\text{\,}\mathrm{M}_{\sun}$
($5\text{\,}\mathrm{M}_{\sun}$) to create the plots in Figure 5 (Figure 6).
The donor star must be more massive than $M_{\mathrm{2}}$. We chose a value
between $M_{\mathrm{2}}$ and $8.4\text{\,}\mathrm{M}_{\sun}$, using a uniform
distribution. Considering that, at the time of RL filling the donor may have
lost some mass, we define $M_{\mathrm{WD}}^{\mathrm{max}}$ to be the largest
possible WD mass, and select $M_{\mathrm{c}}$ to have a value between
$0.1\text{\,}\mathrm{M}_{\sun}$ and $M_{\mathrm{WD}}^{\mathrm{max}}$, using a
uniform distribution.

The donor’s mass and core mass allow us to estimate the radius of the donor
(hence the RL radius) at the time of RL filling. The RL radius and the masses
of the donor and companion provide an estimate of the orbital separation at
the time the CE started. The envelope mass was the difference between the
donor’s mass and the mass of its core. Considering only those binaries in
which the envelope mass as computed here lies within the range defined by the
minimum and maximum expected values of $M_{\mathrm{e}}$, we had all of the
information needed to compute $a(f)$ and $\tau_{\mathrm{merge}}$.

For each of 33 values of $\eta$ in the range 0.5 to 16, we mapped the
evolution of $10^{6}$ binaries through the CE phase. We counted the numbers of
systems that merge within $2\times 10^{5}\text{\,}\mathrm{yrs}$ (red), the
number that merge within a Hubble time (within $1.37\times
10^{10}\text{\,}\mathrm{yrs}$, green), and the number that take longer than a
Hubble time to merge (black). The first category was created to explicitly
consider the proportion of systems that may merge before the CE has a chance
to disperse, since these may exhibit distinctive features. Those that merge
within this short interval or else within a Hubble time, may be used to
compute the rates of potentially observable events.

The top panel in Figures 5 and 6 shows the logarithm to the base 10 of the
number of mergers. Each subsequent panel shows the logarithm to the base 10 of
the average value of one quantity: the time to merger (in years), the mass
ratio $q=M_{\mathrm{c}}/M_{\mathrm{2}}$, the mass ratio
$M_{\mathrm{e}}/M_{\mathrm{c}}$, the mass ratio
$M_{\mathrm{e}}/(M_{\mathrm{c}}+M_{\mathrm{2}})$. To explore the effect of
increasing $\eta$, we can start at the left of each panel and move toward the
right.

Clearly, higher values of $\eta$ produce more frequent and earlier mergers.
The average mass ratio of the post CE binary is not monotonic. Usually, the
core of the donor is always less massive than the companion star. In general,
the envelope masses are larger than the core masses at the onset of our
simulated CE. This even holds when comparing the envelope to the total post CE
binary mass instead of the donor’s core mass. As we fixed the companion mass,
the two lower panels of Figure 5 show the same trend of having lower envelope
masses for larger values of $\eta$. Thus, the cutoff at lower $\eta$ values is
caused by the choice of the maximally allowed envelope mass. Vice versa, the
cuts at higher values of $\eta$ are related to the minimum envelope mass,
which is often given by the requirement that the donor mass should be larger
than the companion mass. By comparing Figures 5, for which
$M_{\mathrm{2}}=$0.8\text{\,}\mathrm{M}_{\sun}$$, and 6, for which
$M_{\mathrm{2}}=$5.0\text{\,}\mathrm{M}_{\sun}$$, find that larger values of
$\eta$ are required to achieve mergers when the companion mass is larger. For
example, there is a limited range for $\eta$ where the three groups according
to the merger time have comparable values, and that this value increases as
$M_{\mathrm{2}}$ increases.

####  4.4.4 Challenges of employing a fixed value of $\eta$

When conducting population synthesis calculations, we start with many
individual binaries and follow the evolution of each. For those binaries that,
at some time during their evolution, start a CE, we must decide what the value
of $a(f)$ will be. For parametric formalisms, including the $\alpha$ formalism
for example, the simulation may be run multiple times, with different CE
parameter values selected for each simulation. Thus, for example, a constant
value of $\alpha\,\lambda$ may be used in each simulation, and the results of
different simulations are compared to test the effects of employing different
values of the CE parameter(s).

If we consider a set of simulations using a single value of $\alpha\,\lambda$,
we know that the value employed is likely to be a good approximation of the
true value for a subset of the binaries. The computed CE final states of these
binaries would be similar to the final states that would be produced in
nature. But for most other binaries, the computed and true final states would
be different. Nevertheless, a set of simulations which employs a different
value for the product $\alpha\,\lambda$ would yield correct results for a
different set of binaries. If we understood the CE phase, we would know which
values of $\alpha\,\lambda$ are appropriate for any given binary. Without this
knowledge, when we use different parameter values for different simulations,
the meaning of the comparisons between simulations is difficult to untangle.
The same is naturally true if we choose to use a single value of $\eta$ in a
specific simulation, and vary the values among simulations.444The same could
in principle be said about employing a fixed value of $\delta$, but, as shown
above, we have selected a regime in which the results should be robust with
respect to modest changes in $\delta.$

The positive feature associated with using a fixed value of the CE parameters
is that it simplifies the input to the simulations. If, however, we had a
simple functional form to express the CE parameters in terms of the binary’s
physical characteristics, we could maintain the simplicity while achieving a
more readily interpreted set of population synthesis results. A formula-based
approach has been taken in other formalisms (De Marco et al., 2011). In the
rest of this paper we explore a trend that allows us to derive and use a
functional form for $\eta.$

##  5 Functional Form for $\eta$

Figure 7: The dependence of $\eta$ on
$M_{\mathrm{e}}/M_{\mathrm{t}}=q_{\mathrm{ec}}q/(1+q)$. Note that $\eta$ is a
function of only $q$ and $q_{\mathrm{ec}}$. The different panels are for
different types of post-CE systems (PCEs) from top to bottom: WD-MS, WD-NS,
double WD. The colors and symbols are the same as in Figure 4. Here each
observed system is represented by three data points using the first, second,
and third quartile between $M_{\mathrm{e}}^{\mathrm{min}}$ and
$M_{\mathrm{e}}^{\mathrm{max}}$ for the envelope mass.

Figure 7 shows $\log_{10}[\eta]$ plotted against
$\log_{10}[M_{\mathrm{e}}/M_{\mathrm{t}}]$. Note that

$\frac{M_{{\mathrm{e}}}}{M_{{\mathrm{t}}}}=\frac{q_{\mathrm{ec}}q}{1+q},$ (21)

where $M_{\mathrm{t}}=M_{\mathrm{c}}+M_{\mathrm{2}}$. Thus, this graph shows
$\eta$ as a function of a variable involving only $q$ and $q_{\mathrm{ec}}$.
In the top panel, the points correspond to WD-MS post-CE binaries. The points
exhibit a linear trend, which is also consistent with the more sparsely
populated middle and bottom panels, showing WD-NS and WD-WD systems,
respectively.

Each system here is represented by three data points corresponding to three
choices of the envelope mass. To select the values for $M_{\mathrm{e}}$, we
computed the maximum and minimum envelope mass (as in the previous subsection)
for each donor star. We then divided this interval into four equal pieces, and
selected three values of $M_{\mathrm{e}}$: the middle value and the ones just
above and below the middle value. Thus, we excluded only the minimum and
maximum possible values of the envelope mass. The physical system may have had
an envelope mass close to one of these values, but not necessarily. If we knew
the actual value of the envelope mass for each system, we would be able to re-
plot the figure and determine the most appropriate functional form. Without
the ability to do that, we can use the points in Figure 7 as a guide to the
trend that expresses the functional form in an approximate way as follows:

$\log_{10}[\eta]=-A\,\log_{10}[M_{\mathrm{e}}/M_{\mathrm{t}}]+B.$ (22)

In the case of $A=0$, $\eta$ becomes constant independent of binary
parameters. To derive the fit parameters ($A$ and $B$) we employ a straight
line model. Given the data at hand, we cannot justify a more complicated
model. Whether or not a straight-line fit is the optimal choice, the fact some
functional form links $\eta$ to a set of binary-system parameters is
important. It allows us to eliminate $\eta$ from the equations for
$a(0)/a(f)$. For each set of post-CE binaries, the ratio of initial to final
separations can be expressed as a function of $q$ and $q_{\mathrm{ec}}$. BPS
calculations can compute post-CE separation analytically. This simplification
makes the formalism relatively straightforward to implement, an important
consideration for BPS calculations.

We now have

$\frac{a(f)}{a(0)}=\left[\frac{\left(1+q\right)\left(1+q_{\mathrm{ec}}\right)^{2}}{1+q\,\left(1+q_{\mathrm{ec}}\right)}\right]\,e^{\left[-2\mathcal{F}(q,\delta)\,\left(10^{B+\Delta}\times\left(\frac{q_{\mathrm{ec}}q}{1+q}\right)^{1-A}\right)\right]}.$
(23)

We see that only the values of $q$ and $q_{\mathrm{ec}}$ are needed in order
to compute $a(f)/a(0)$. There is no explicit dependence on $\eta$, which has
been replaced by it’s functional form. The values of $A$ and $B$ are
determined by the type of binary we are considering. We see that larger values
of $B$ produce smaller final orbits. Variations in $A$ can produce complex
behaviors. For $A=1,$ the exponent has no dependence on $M_{\mathrm{e}}$. For
$A<1$, orbital shrinkage is facilitated as $q_{\mathrm{ec}}$ and as $A$
decreases. For $A>1$ orbital shrinkage is moderated as $A$ increases and also
as $q_{\mathrm{ec}}$ increases. In the expression on the right-hand-side, we
have added a term $\Delta$ to $B$. When conducting simulations, we can use
this term to generate the spread associated with the residuals by selecting
$\Delta$ from a Gaussian distribution with width $\sigma.$

Fits to the data in Figure 7 produce values for the parameters $A$ and $B$
that are given in Table 2. Although $\sigma$, also in the table and derived
from the distribution of residuals, provides a rough estimate of the
uncertainties, a deeper source of uncertainty is that we do not know which
choice of the envelope mass is best. In separate work we consider possible
scenarios in which the envelope masses and parameters can be chosen in a
mutually consistent way.

Table 2 shows the results for the combined data (“All”, top row), as well as
the fits for six types of post-CE binary. Note that, in this row, $A\approx 1$
and $B\approx 0.6$. These values are close to those found in all cases with
small to modest shrinkage (${\cal R}_{\mathrm{d}}<20$; rows 3, 5, and 7 in
Table 2).

Table 2:  Functional form of $\eta$: fits of Equation (22) to plots in Figure 7. The last column is the approximated standard deviation of the residuals. Type |  Slope, $A$ |  Intercept, $B$ | $1\,\sigma$
---|---|---|---
All | $0.952\pm 0.011$ | $0.603\pm 0.006$ | $0.08$
WD+MS (Shrink $>20$) | $0.892\pm 0.021$ | $0.567\pm 0.014$ | $0.08$
WD+MS (Shrink $<20$) | $1.027\pm 0.047$ | $0.637\pm 0.008$ | $0.08$
WD+NS (Shrink $>20$) | $1.285\pm 0.072$ | $0.678\pm 0.024$ | $0.08$
WD+NS (Shrink $<20$) | $1.051\pm 0.120$ | $0.561\pm 0.020$ | $0.11$
WD+WD (Shrink $>20$) | $0.780\pm 0.046$ | $0.487\pm 0.027$ | $0.065$
WD+WD (Shrink $<20$) | $0.969\pm 0.130$ | $0.640\pm 0.034$ | $0.03$
Figure 8: In the new formalism, the ratio $a(f)/a(0)$ is shown as a function
of the final mass ratio $q=M_{\mathrm{c}}/M_{\mathrm{2}}$. The only other
variable is $q_{\mathrm{ec}}=M_{\mathrm{e}}/M_{\mathrm{c}}$, which is sampled
by the separate curves as indicated in the bottom panel. The points in each
panel represent a particular set of the post-CE binaries; the top three panels
are for double WD, WD-NS, and WD-MS, with a substantial shrinkage, cf. blue
and green points in Figure 4. The bottom panel only contains binaries with
$\mathcal{R}_{\mathrm{d}}<20$, cf. red points in Figure 4.

Figure 8 plots curves of $\log_{10}[a(f)/a(0)]$ versus $q$, for each of nine
values of $q_{\mathrm{ec}}$ (color-coded), given in the bottom panel. We also
show points superposed on each panel. These correspond to the post-CE binaries
most likely to have actually experienced a significant CE episode.

In the bottom panel we use $A=1$ and $B=0.6$. Table 2 shows that, whatever the
composition of the binary, the systems that have had little shrinkage
($\mathcal{R}_{\mathrm{d}}<20$) have values of $A$ and $B$ close to $1.0$ and
$0.6$, respectively. The curves in this panel are non-intersecting. Larger
values of $q_{\mathrm{ec}}$, which must at least sometimes correspond to large
$M_{\mathrm{e}}$, exhibit less shrinkage, and some even expand. The points
superposed are the post-CE binaries that have ($\mathcal{R}_{\mathrm{d}}<20$);
they include WD-MS, WD-NS, and WD-WD binaries, drawn from rows 3, 5, and 7 of
Table 2. . This panel demonstrates that, because SCATTER’s physical
assumptions are closely tied to angular momentum conservation, modest orbital
shrinkage, and even orbital expansion, are covered by the SCATTER formalism.
The fact that these types of post-CE candidates have positions encompassed by
the curves in this panel may indicate that there is a relatively smooth
transition between some CE systems and binaries experiencing stable RL
overflow.

Row 2 in the table corresponds to WD-MS binaries with ${\cal
R}_{{\mathrm{d}}}>20$. These are likely CE survivors. Recall that WD-MS
binaries are post-CE binaries with a WD in orbit with a main-sequence star.
The pre-CE system to which the equation applies is a MS-giant binary. Rows 4
and 6 also show binaries with ${\cal R}_{{\mathrm{d}}}>20$, but for WD-NS and
WD-WD binaries, respectively. The values of $A$ and $B$ are different for each
type of post-CE binary.

The second panel from the bottom in Figure 8 panel shows WD-MS post-CE
binaries with $\mathcal{R}_{\mathrm{d}}>20$. This panel applies to the largest
number of post-CE binaries in any single category, shown superposed in the
analytic curves. We see that for $A<1$, there can be expansion of the CE
binary for low values of $q$, and that the expansion is associated with high
envelope mass. As $q$ increases, however, the curves cross and that the
binaries which shrink most have large values of $M_{\mathrm{e}}$. Most of the
data points fit well within the region where shrinkage is expected, with
$a(f)/a(0)$ going down as low as $10^{-3}$. A few points are just outside the
curves, and seem to exhibit even more shrinkage. Thus may indicate that the
value of $A$ and/or $B$ found through the fit is too small, or it may simply
reflect our uncertainty.

The top two panels correspond to binary types that were not well represented
in our set of systems: WD-NS systems in the third panel from the bottom and
WD-WD systems in the top panel. Given the small numbers of WD-WD and WD-NS
systems, it is difficult to quantify the uncertainty. We have used $\Delta$,
which in this paper we have taken simply to be $\sigma$, but as we point out
in § 6, the true uncertainties are likely to be larger.

##  6 Application

Figure 9: For WD-WD binaries, times to merger through gravitational radiation
after the CE, plotted against the pre-CE time to merger, the total binary
mass, and the mass of the donor’s core. The parameter values are shown in the
figure and described in the text. Green: the parameters used are $A$ and $B$
from the middle panel in the row. Blue: $B$ is replaced by $B+\Delta$ with
$\Delta=1\,\sigma.$ Red: $B$ is replaced by $B-\Delta$ with
$\Delta=1\,\sigma.$

As discussed in the introduction, SNe Ia progenitor models generally involve
binaries which undergo one or two CE phases. In the case of two phases, the
second CE typically leads to a double-WD state. If the two WDs come close
enough together that one of them will fill its RL, there could be a merger.
Several types of mergers (e.g., both CO-CO mergers and CO-He mergers) may be
able to produce SNe Ia. The time to merger after the CE phase can be estimated
by using equation (17).

The goal of this paper is not to assess the likelihood of mergers within a
realistic stellar population, but instead to study how the SCATTER formalism
works in producing close WDs, and to develop a better understanding of the
uncertainties. To do this, we generated binaries containing a single WD and a
companion star that fills its RL at a point when mass transfer will be
dynamically unstable. The RL filling condition determines the pre-CE orbital
separation. We started with a WD of $0.8\text{\,}\mathrm{M}_{\sun}$, and then
introduced a more massive companion
($$0.8\text{\,}\mathrm{M}_{\sun}$<M_{\mathrm{1}}<$8.4\text{\,}\mathrm{M}_{\sun}$$
with an evolved core ($M_{\mathrm{c}}>$0.1\text{\,}\mathrm{M}_{\sun}$$). We
chose the mass values from uniform distributions.

Starting with the initial states described above, we used the SCATTER
formalism to compute the post-CE separation between the donor’s core, which
itself becomes a WD, and the first-formed WD. Thus, the resulting system is
comprised of two WDs in a binary whose orbital radius we have computed. We
have enough information to compute the time to merger, assuming that the post-
CE binary has a circular orbit. The results are shown in Figure 9. We do not
make any assumptions about which pairs of WDs can produce SNe Ia. (See Soker
2019 and Ruiter 2020 for a summary of possible channels.)

The primary goal of this exercise is to study the uncertainties, which are
related to our uncertainties in the values of $A$ and $B$. In particular,
because there were relatively few WD-WD systems among the post-CE binaries in
our sample, the values of $A$ and $B$ are more uncertain than for WD-MS
binaries. Additionally, when considering the observed WD-WD post-CE
candidates, we did not know which WD had evolved first, and assumed that it
was the more massive WD; this may not in fact have been the case for all
binaries in our sample. We therefore used two sets of values for the pair
$(A,B)$, as shown in the panels of Figure 9.

In the top panels we used values close to the average for the entire sample
($A=0.952,B=0.603)$, with $\sigma=0.08$. In the bottom panels we use the
values derived specifically for WD-WD binaries ($A=0.780,B=0.487)$, with
$\sigma=0.065$. Mergers taking place within a Hubble time are represented by
systems lying below the black line in each panel. The parameters we derived
based on WD-WD post-CE binaries produced fewer mergers within a Hubble time.

Even if we have a well-sampled group of binaries for the WD-WD model, there is
still an uncertainty based on the residuals to the fit. Thus, when we compute
$A$ and $B$ for a given system that will become a close double-WD binary, we
must take into account the uncertainty $\sigma.$ In the context of a
population synthesis simulation, for a given system, $i$, we would take:
$B=B+\kappa\times\sigma$, where we choose $\kappa$ from a Gaussian
distribution of width $\sigma$. Thus, even for a single binary in the
simulation, a range of possible values of $a(f)$ are possible. This
uncertainty is derived based on the actual post-CE binaries with which we
started. It reflects a genuine and expected uncertainty due to the fact that,
even if the variables we have used for two different binaries are identical,
there will be some physical differences between the binaries that should
produce a spread in the values of $a(f).$

To explore the differences, we consider, for each pair $(A,B)$, what happens
if we use $B+\Delta$, where we have selected $\Delta=\sigma$ and also
$B-\Delta$. We find that larger effective values of $B$ always produce smaller
final-state binaries, as the formula indicates they should.

We will show in the next section that the changes in separation predicted by
the SCATTER formalism can be modest, compared with those predicted by other
formalisms. It is therefore important to note that in each case, a significant
number of mergers are expected within a Hubble time. In the upper (lower)
panels $100\%$ ($74\%$) of all binaries for which $B$ is replaced by
$B+\Delta$ merged in a Hubble time. We would expect roughly a third of all
binaries in the simulations to have low values of $B$. In addition, some
binaries ($52\%$ in the upper panels, $10\%$ in the lower panels) also merged.

In the bottom, taken from the WD-WD post-CE binaries, the mergers tended to
have a lower-mass WD (possibly a helium WD), merging with a WD of higher mass.
In the upper panel, the masses of the merging WDs were on average higher. Both
situations provide potential channels to SNe Ia.

##  7 Comparisons with other Formalisms

###  7.1 The $\alpha$ Formalism

The so-called $\alpha$ formalism (Webbink, 1984; de Kool, 1990) is a
traditional and more common way the deal with a CE. It is based on energy
conservation. In its original form it assumed that orbital energy is the only
source providing the energy needed to unbind the envelope material. The
formalism contains two parameters: $\alpha$ describes the efficiency of the
energy conversion; $\lambda$ describes the strength of envelope binding. In
the final equation, which describes the change of the orbital separation,
those two parameters appear only in a product, so that some studies employ the
product as the only free parameter.

$\frac{a(0)}{a(f)}=\left(1+q_{\mathrm{ec}}\right)\,\left(1+\frac{2\,q_{\mathrm{ec}}\,q}{\alpha\,\lambda}\,\left[f(q+q_{\mathrm{ec}}\,q)\right]^{-1}\right).$
(24)

In Figure 10 we compare the SCATTER formalism with the $\alpha$ formalism,
using three values of the product $\alpha\,\lambda$ sometimes applied in
simulations: $\alpha\,\lambda=0.1,1,10$. There are several differences between
the formalisms.

First, the SCATTER formalism (in common with the angular-momentum based
$\gamma$ formalism discussed below), allows for orbital expansion, as well as
for the shrinking of the orbit. This is because orbital evolution during the
CE phase depends on the amount of angular momentum carried by mass exiting the
system.

Figure 10: In both the $\alpha$ formalism (varying $\alpha\,\lambda$) and the
SCATTER formalism for post-CE systems, the ratio $a(f)/a(0)$ is shown as a
function of the final mass ratio $q=M_{\mathrm{c}}/M_{\mathrm{2}}$. The only
other variable is $q_{\mathrm{ec}}=M_{\mathrm{e}}/M_{\mathrm{c}}$, which is
sampled by the separate curves as shown in the bottom panel of Figure 8.
Figure 11: In both the $\gamma$ formalism and the new formalism, the ratio
$a(f)/a(0)$ is shown as a function of the final mass ratio
$q=M_{\mathrm{c}}/M_{\mathrm{2}}$. The only other variable is
$q_{\mathrm{ec}}=M_{\mathrm{e}}/M_{\mathrm{c}}$, which is samples by the
separate curves as shown in the bottom panel of Figure 8.

Second, the shrinkage is a monotonic function of $q$ in the $\alpha$
formalism, and not in the SCATTER formalism, where the maximum amount of
shrinkage occurs at $q=1.$ In addition, we note that to conduct a full
comparison with the SCATTER formalism, a range of different values of $A$ and
$B$ must be considered. If we refer to Figure 8, it becomes clear that, while
shrinkage in the $\alpha$ formalism always increases with increasing envelope
mass, the same is not true of the SCATTER formalism, as we have already
discussed.

Finally, we see that the $\alpha$ formalism generally produces more shrinkage.
We might therefore expect it to yield more mergers within the CE, while fewer
may occur in the SCATTER formalism. Thus, SCATTER may allow more systems to
survive a first CE epoch. It may also lead to merger events that occur at
later times. If predictions of similar types are made by detailed BPS
simulations, these comparisons will be testable.

###  7.2 The $\gamma$ Formalism

In the $\gamma$ formalism (Nelemans et al., 2000; Nelemans & Tout, 2005) the
CE is ejected because of angular momentum imparted by the binary. It was
invented to explain post CE binaries with a mild or even no orbital shrinkage.
The $\gamma$ formalism has only one free parameter which describes the
strength of the proportionality between angular momentum and mass change. The
change of the orbital separation can be described by

$\frac{a(0)}{a(f)}=\left[\frac{1}{1+q_{\mathrm{ec}}}\right]^{2}\,\left[\frac{1+q\,(1+q_{\mathrm{ec}})}{1+q}\right]\,\left[\frac{1+q\,(1+q_{\mathrm{ec}})}{1+q\,+q\,q_{\mathrm{ec}}\,(1-\gamma)}\right]^{2}.$
(25)

As in the SCATTER formalism, this angular-momentum based approach allows some
orbits to experience only modest shrinkage, or even expansion. On the other
hand, there are sharply defined regions of the parameter space that are
associated with dramatic orbital shrinkage. In Figure 11 this fine tuning is
visible by the deep dips which shift their position depending on the value of
$\gamma$. The SCATTER formalism does not exhibit such singular behavior.

In this regard the SCATTER formalism can reproduce the main required features
of both existing formalisms. At the same time the equation, which describes
the orbital change between the initial and final state remains on a similar
complexity. This allows an easy usage in any kind of code which needs to deal
with CE evolution.

##  8 Conclusions

The reason the CE plays such an important role in the study of binary systems
is that a significant fraction of binaries experience a CE. Most important,
the CE end state determines whether a new phase of mass transfer or merger
follows. Being able to predict CE outcomes is crucial to correct computations
of many event rates, such as the rates of SNe Ia or of gravitational mergers.
As we have seen in this work however, and in fact common to all CE formalisms,
there are unavoidable uncertainties. Fortunately, the key question is not
whether we can make an exact prediction for an individual system. Rather it is
whether, when we start with a full population of binaries, do the results
match observations of binary populations?

The predictions of existing approaches to CE parameterization have not been
able to match the full complement of present-day observations. It will be
important to use SCATTER to make many sets of predictions and to check them
against observations. As with all formalisms, checks can be performed in
astronomical contexts ranging from cataclysmic variables to high-mass X-ray
binaries. The delay-time distributions for SNe Ia and measures of chemical
enhancement can both provide checks, as can the rates of BH-BH, BH-NS, and NS-
NS mergers. Although we don’t know yet how well SCATTER will fare in these
tests, its approach has the advantage that, at its heart, is simple physics
that we know applies. Furthermore the formalism is complex enough, in
comparison with the standard $\gamma$ formalism, to be flexible.

Our original motivation in developing SCATTER was to find a natural way to
compute CE end states for hierarchical triples and other higher-order
multiples. By focusing on the interaction of “Single Components” with the CE,
one can develop natural generalizations in which there are more than two
components. The fact that a simple functional form for the parameters can
emerge from the formalism is a bonus associated with the formalism’s use of
angular momentum.

Generalizations in addition to the application to higher-order multiples are
also to be expected. Of most interest perhaps is the possibility of using more
than one component of the angular momentum. The mathematical advantages that
would provide are that computations for more complex systems may be possible.
From a physical perspective, when the components of the binary (or higher-
order multiple) can impart angular momentum in all three directions, the end-
state can generally be expected to spin, leading to predictions that can be
tested.

SCATTER suggests several directions for future research. One type of
additional work will be the application of the formalism to cases not
considered in this paper. The basic principles discussed in Section 3 apply to
a wide range of systems, the phenomenological approach we have employed
allowed us to derive functional parameters for only the types of systems
identified as post-CE binaries in which a WD has a MS, WD, or NS companion.
Future investigations will consider planetary systems and, at the other end of
the mass spectrum, black holes. We expect that this is just the first
implementation of the SCATTER formalism and that its basic principles will
prove useful in predicting the evolution of systems containing different types
of stars, including those with high multiplicities.

## Acknowledgements

RD acknowledges support from the National Science Foundation, through NSF
AST-2009520 and NSF PHY-1748958. She also acknowledges useful discussions with
Ilya Mandel and Ashley Ruiter. MK was partly supported by Grant No 12090040,
and 12090043 of the Natural Science Foundation of China. MK acknowledges
support from the Swiss National Science Foundation Professorship Grant (PP00P2
211006; PI Fragos). YG is funded by the Royal Society and the K.C. Wong
Education Foundation. CK acknowledges funding from the UK Science and
Technology Facility Council (STFC) through grant ST/R000905/1 and
ST/V000632/1, and a travel support from Z. Han. This team collaboration
started at a conference “Progenitors of Type Ia Supernovae” in Lijiang, 2019.
We thank the referee, Noam Soker, for helpful comments.

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##  Appendix A The minimum and maximum envelope mass

Figure 12: Helium core mass vs. envelope mass for stellar models with initial
masses up to $10\text{\,}\mathrm{M}_{\sun}$ and an age below the age of the
Universe. Top: models calculated with BEC at $Z=0.0088$ (Kruckow et al.,
2018). The colour shows the initial stellar mass. The think lines with the
pluses are the phases of stellar expansion and the thin lines with the crosses
are phases of contraction.

Figure 13: Like Figure 4 but showing WD-WD binaries (left set of panels) and
WD+NS binaries (right set of panels).

We have used the stellar models presented in Kruckow et al. (2018) (calculated
with BEC, $Z=0.0088$) to determine a minimum and a maximum mass for the
envelope above the helium rich core. From those models we get two rough
descriptions for the envelope mass range given a core mass, see Fig. 12.

$\displaystyle M_{\mathrm{e}}^{\mathrm{min}}$ $\displaystyle=$
$\displaystyle\begin{cases}$0.9\text{\,}\mathrm{M}_{\sun}$-0.85\,M_{\mathrm{c}}&\text{if
}M_{\mathrm{c}}\leq$0.5\text{\,}\mathrm{M}_{\sun}$,\\\
$0.8\text{\,}\mathrm{M}_{\sun}$+2.1\,M_{\mathrm{c}}&\text{otherwise},\end{cases}$
(A1) $\displaystyle M_{\mathrm{e}}^{\mathrm{max}}$ $\displaystyle=$
$\displaystyle$1.3\text{\,}\mathrm{M}_{\sun}$+10.5\,M_{\mathrm{c}}.$ (A2)

When using the equations above we used additional limitations, e.g., by
setting an additional solid limit on the minimum and maximum, see § 4.4.

##  Appendix B Binaries with a White Dwarf and Compact Companion

Table 1 indicates that we had both WD-WD and WD-NS post-CE candidate binaries.
We considered these in a manner exactly analogous to the way we considered WD-
MS binaries. Figure B.1 shows for WD-WD binaries (top set of panels) and for
WD-NS binaries (bottom set of panels) the same set of graphs as shown in
Figure 3. The primary difference in these calculations is that Star 2 is a
compact object. There is therefore no opportunity for a second epoch of mass
transfer555If gravitational radiation brings the two compact objects close
enough to each other that one fills its Roche lobe, these binaries could
represent rare systems, such as AM CVn stars in which a compact object donates
mass to compact companion..

Unfortunately, double-compact binaries are dim and more challenging to
identify than binaries consisting of a compact object in orbit with an
extended star. we therefore have fewer examples of them. Patterns easily
discernible in the WD-MS systems are more difficult to identify. For CE end
states, the most important feature is that there is no single value, or
limited range of values, of $\eta$ associated with these sets of binaries.
Fortunately though, as we saw in Figure 7, the WD-WD and WD-NS binaries are
consistent with the same functional form we derived for WD-MS binaries. That
is, $\eta$ can be written as $A\,\log_{10}[M_{\mathrm{e}}/M_{\mathrm{t}}]+B$.