ORBITAL EVOLUTION OF MASS-TRANSFERRING ECCENTRIC BINARY SYSTEMS. II. SECULAR EVOLUTION

The evolution of stellar binaries can be affected by a variety of physical phenomena, including mass loss and mass transfer processes. Such processes affect the orbital evolution of these systems, and thus constitute an important part of the binary evolution theory. In Dosopoulou & Kalogera (2016, hereafter Paper I), we treated different cases of mass loss/transfer as perturbations to the general two-body problem. In that paper, we described the general perturbation method that can be used to calculate the orbital evolution of the binary and using this method we derived the general phase-dependent time-evolution equations for the semimajor axis and the eccentricity. Here, assuming that the binary is slowly evolving, we orbit-average the phase-dependent time-evolution equations and derive the secular time-evolution equations for the semimajor axis and the eccentricity. The secular evolution equations are derived both in the adiabatic regime as well as under the assumption of delta-function perturbation.

We consider two bodies of masses M₁ and M₂ in an eccentric binary system of total mass M. We refer to these as the donor and the accretor, respectively. The donor M₁ is losing mass with a rate ${\dot{M}}_{1}$ and the accretor is accreting mass with a rate ${\dot{M}}_{2}$, so that the total mass loss rate is given by $\dot{M}={\dot{M}}_{1}+{\dot{M}}_{2}$. For non-conservative mass transfer, only a fraction γ of the lost mass is accreted. This means that in this case, we have ${\dot{M}}_{2}=-\gamma {\dot{M}}_{1}$, where ${\dot{M}}_{1}\lt 0$. For the general case of non-conservative mass transfer, in Paper I we derived the following phase-dependent time-evolution equations of the osculating semimajor axis a and eccentricity e in the K_R reference system (see Figure 1 here and Equations (117) and (118) in Paper I):

$$\begin{eqnarray}{\left(\displaystyle \frac{{da}}{{dt}}\right)}_{\mathrm{non} \mbox{-} \mathrm{con}} & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[{b}_{r}e\mathrm{sin}\nu +{b}_{\tau }(1+e\mathrm{cos}\nu )]\\ & & -2{ha}\left[\displaystyle \frac{{e}^{2}+2e\mathrm{cos}\nu +1}{1-{e}^{2}}\right]\\ & & -(\zeta +1)\left[\displaystyle \frac{{e}^{2}+2e\mathrm{cos}\nu +1}{1-{e}^{2}}\right]\displaystyle \frac{\dot{M}}{M}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}{\left(\displaystyle \frac{{de}}{{dt}}\right)}_{\mathrm{non} \mbox{-} \mathrm{con}} & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[{b}_{r}\mathrm{sin}\nu \\ & & \left.+{b}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right]\\ & & -2h[e+\mathrm{cos}\nu ]-(\zeta +1)[e+\mathrm{cos}\nu ]\displaystyle \frac{\dot{M}}{M},\end{eqnarray} \tag{ 2 }$$

where the subscripts r and τ refer to the radial and tangential components, respectively, ν is the true anomaly, and ζ is the fraction of the total orbital angular momentum removed from the system. The other parameters are defined by

$$\begin{eqnarray}{\boldsymbol{b}} & \equiv & {\dot{M}}_{2}\left(\displaystyle \frac{{{\boldsymbol{w}}}_{2}}{{M}_{2}}+\displaystyle \frac{{\omega }_{\mathrm{orb}}\times {{\boldsymbol{r}}}_{{A}_{2}}}{{M}_{2}}+\displaystyle \frac{{{\boldsymbol{w}}}_{1}}{{M}_{1}}+\displaystyle \frac{{\omega }_{\mathrm{orb}}\times {{\boldsymbol{r}}}_{{A}_{1}}}{{M}_{1}}\right)\\ & & +{\ddot{M}}_{2}\left(\displaystyle \frac{{{\boldsymbol{r}}}_{{A}_{2}}}{{M}_{2}}+\displaystyle \frac{{{\boldsymbol{r}}}_{{A}_{1}}}{{M}_{1}}\right)\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&h\equiv {\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right)\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\dot{M}}_{2}={\dot{M}}_{2,\mathrm{acc}}=-\gamma {\dot{M}}_{1}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\dot{M}}_{1}=-{\dot{M}}_{2,\mathrm{acc}}={\dot{M}}_{1,\mathrm{acc}}=\gamma {\dot{M}}_{1}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\dot{M}=(1-\gamma ){\dot{M}}_{1}.\end{eqnarray} \tag{ 7 }$$

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In Equations (3)–(7), ${{\boldsymbol{w}}}_{1}$ and ${{\boldsymbol{w}}}_{2}$ are the absolute velocities of the ejected and accreted mass, respectively, and ${{\boldsymbol{r}}}_{{A}_{1}}$ and ${{\boldsymbol{r}}}_{{A}_{1}}$ are the positions of the ejection and accretion points relative to the bodies' centers of mass. Furthermore, ${{\boldsymbol{v}}}_{1}$ and ${{\boldsymbol{v}}}_{2}$ are the orbital velocities of the two bodies with respect to the inertial frame and ${\omega }_{\mathrm{orb}}$ is the orbital angular frequency.

In the case the bodies in the binary system can be treated as point-masses, we can simplify Equations (1) and (2) and rewrite them as (see Equations (123) and (124) in Paper I)

$$\begin{eqnarray}{\left(\displaystyle \frac{{da}}{{dt}}\right)}_{\mathrm{non} \mbox{-} \mathrm{con}}^{\mathrm{point}} & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[{c}_{r}e\mathrm{sin}\nu +{c}_{\tau }(1+e\mathrm{cos}\nu )]\\ & & -(\zeta +1)\left[\displaystyle \frac{{e}^{2}+2e\mathrm{cos}\nu +1}{1-{e}^{2}}\right]\displaystyle \frac{\dot{M}}{M}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}{\left(\displaystyle \frac{{de}}{{dt}}\right)}_{\mathrm{non} \mbox{-} \mathrm{con}}^{\mathrm{point}} & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[{c}_{r}\mathrm{sin}\nu \\ & & \left.+{c}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right]\\ & & -(\zeta +1)[e+\mathrm{cos}\nu ]\displaystyle \frac{\dot{M}}{M},\end{eqnarray} \tag{ 9 }$$

where we make use of the definition

$$\begin{eqnarray}&&{\boldsymbol{c}}\equiv \displaystyle \frac{\dot{{M}_{2}}}{{M}_{2}}({{\boldsymbol{w}}}_{2}-{{\boldsymbol{v}}}_{2})-\displaystyle \frac{\dot{{M}_{1}}}{{M}_{1}}({{\boldsymbol{w}}}_{1}-{{\boldsymbol{v}}}_{1})\end{eqnarray} \tag{ 10 }$$

and the subscripts r and τ refer to the radial and tangential components, respectively.

Equations (1) and (2) are generally phase-dependent and indicate that the semimajor axis and the eccentricity change over orbital timescales undergoing periodic oscillations. This makes these equations difficult to implement in binary evolution codes like StarTrack (Belczynski et al. 2008), BSE (Hurley et al. 2002), or MESA (Paxton et al. 2011, 2013, 2015). This difficulty emerges from the fact that the semimajor axis and eccentricity change on an orbital timescale ${\tau }_{\mathrm{orb}}$, while in the stellar evolution codes mentioned before the time-step is limited by the thermal timescale ${\tau }_{\mathrm{th}}$, where ${\tau }_{\mathrm{th}}\gg {\tau }_{\mathrm{orb}}$. This means that these codes cannot keep track of the evolutionary changes in the orbital elements using Equations (1) and (2). However, Equations (1) and (2) can be simplified if they are orbit-averaged—a procedure that can smooth out the aforementioned oscillations. When the assumption of slowly evolving systems can be made, the orbit-averaged equations provide information about the secular evolution of the orbital elements, which from now on we will refer to as secular time-evolution equations. The secular time-evolution equations can then be implemented in the stellar evolution codes mentioned before. In this paper, we perform the orbit-averaging process either under the adiabatic approximation or considering delta-function mass loss/transfer at periastron. Under either of these assumptions, the derived secular time-evolution equations for the semimajor axis and the eccentricity are simplified further and can be used to efficiently model interacting eccentric binary systems.

This paper is organized as follows. In Section 2, we present the orbit-averaging process and describe as well as comment on the adiabatic approximation and the assumption of delta-function perturbation. In Section 3, we discuss briefly the importance of the reference frame choice and derive for two different reference frames the secular orbital element time-evolution equations both in the adiabatic regime and for delta-function perturbation under the additional assumption of a phase-independent perturbation. In Section 4, we derive the secular time-evolution equations for the semimajor axis and the eccentricity both in the adiabatic regime and under the assumption of delta-function mass loss/transfer at periastron for the case of isotropic wind mass loss. In Section 5, we apply the results of Section 3 in the case of anisotropic wind mass loss and discuss qualitatively the effect of some simple wind structures on the binary orbital evolution, including jets. In Section 6, we study the non-isotropic ejection and accretion in a binary for the cases of both conservative and non-conservative mass transfer (including Roche-Lobe-Overflow (RLOF)) assuming extended bodies. In Section 7, we derive the secular orbital element time-evolution equations for some commonly used characteristic examples of mass loss/transfer under the additional point-mass approximation. Section 8 deals with the case of a phase-dependent mass accretion rate (Bondi–Hoyle scenario), and in Section 9 we summarize our results.

The time-evolving orbital elements of a perturbed two-body system undergo physical oscillations due to the periodicity of the orbit (see Equations (1) and (2)). These oscillations can be smoothed out by adopting orbit-averaging techniques that provide information about the secular evolution of the system. Following Lagrange planetary equations (which refer to osculating orbital elements) for a general perturbation (see Equations (18)–(23) in Paper I), the time-evolution equation for the true anomaly ν is given by

$$\begin{eqnarray}&&\displaystyle \frac{d\nu }{{dt}}={\left(\displaystyle \frac{d\nu }{{dt}}\right)}_{\mathrm{unperturbed}}+{\left(\displaystyle \frac{d\nu }{{dt}}\right)}_{\mathrm{perturbation}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&=\;\displaystyle \frac{n{(1+e\mathrm{cos}\nu )}^{2}}{{(1-{e}^{2})}^{3/2}}-\displaystyle \frac{d\omega }{{dt}}-\mathrm{cos}i\displaystyle \frac{d{\rm{\Omega }}}{{dt}},\end{eqnarray} \tag{ 12 }$$

where ω is the argument of periapsis and Ω is the longitude of the ascending node. For a general perturbation, the osculating orbit is precessing relative to the unperturbed orbit (see Equations (21)–(23) in Paper I) and the true anomaly time-evolution deviates from the first unperturbed term in Equation (12). The additional terms come from the fact that while time t is measured from a fixed moment in time, ν is measured from the periapsis, which changes through both precessions $\dot{\omega }$ and $\dot{{\rm{\Omega }}}$. For the perturbations induced by mass loss/transfer processes that we study in this paper, we proved in Section 3 in Paper I (see Equation (65) in Paper I) that $\dot{{\rm{\Omega }}}=0$. In addition, assuming that in first-order approximation the precession of the argument of periapsis ω due to mass loss/transfer is small (i.e., $\dot{\omega }\ll 1$), then to first order we can compute the secular evolution of the orbital elements neglecting the second and third terms in Equation (12) and using instead the substitution

$$\begin{eqnarray}&&d\nu =n\displaystyle \frac{{(1+e\mathrm{cos}\nu )}^{2}}{{(1-{e}^{2})}^{3/2}}{dt}\end{eqnarray} \tag{ 13 }$$

when we integrate the equations da/dt and de/dt over ν from 0 to $2\pi$. In order to incorporate higher-order effects in the perturbation equations, we have to proceed one step further and include all of the terms in Equation (12) when applying the orbit-averaging process. In this paper, we derive first-order secular time-evolution equations where the general orbit-averaging rule is defined by the orbit-averaging integral

$$\begin{eqnarray}&&\langle (...)\rangle =\displaystyle \frac{{(1-{e}^{2})}^{3/2}}{2\pi }{\int }_{0}^{2\pi }\ (...)\ \displaystyle \frac{d\nu }{{(1+e\mathrm{cos}\nu )}^{2}},\end{eqnarray} \tag{ 14 }$$

where we made use only of the first term in Equation (12) and the quantity to be orbit-averaged is depicted with the three dots and is generally phase-dependent. In most cases, this makes the above integral tedious and not easy to resolve into a simple analytical form. However, there are cases that lead to significant simplifications to the time-evolution equations for the secular orbital elements. Two commonly used simplifying approximations are the adiabatic approximation and the assumption of a delta-function perturbation that we will describe in detail in the following two subsections. In the next sections, we orbit-average the various phase-dependent time-evolution equations derived in Paper I under both of these approximations.

2.1. Adiabatic Approximation

2.2. Delta-function Perturbation

Under certain conditions, the effect of a perturbation on a two-body system can have its peak at specific orbital phases. In this case, the perturbation force can be approximated like a delta-function perturbation centered at specific points in the orbit.

In the case of mass loss/transfer in eccentric orbits, the evolution of the mass transfer rate has a Gaussian-like behavior, with a maximum value occurring at periastron (e.g., Lajoie & Sills 2011; Davis et al. 2013). Thus, it is reasonable to assume that mass loss will mostly occur near periastron, since this is the point in an eccentric orbit where the component stars are closest to each other. Because of this, it is logical to examine the case of delta-function mass loss/transfer at periastron. In this case, we can assume a mass loss rate ${\dot{M}}_{0}$ averaged over $2\pi$ such that

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{\dot{{M}_{0}}}{2\pi }\ \ \delta (\nu ).\end{eqnarray} \tag{ 15 }$$

Here, M is the mass of the mass-losing/accreting star and $\dot{M}$ is the relevant mass loss/accretion rate. The function $\delta (\nu )$ is the delta-function as a function of the true anomaly ν centered at the periastron position where $\nu =0$.

The time-evolution equations for the osculating orbital elements in their vector form (see Equations (18)–(23) in Paper I) are independent of the reference frame choice. Decomposing the perturbing force into its components in different coordinate systems is helpful and leads to different sets of equations for the semimajor axis and the eccentricity in each system.

In Section 3 of Paper I, we presented the form of the phase-dependent time-evolution equations for the osculating orbital elements in three different reference systems (see Figure 1 here and Section 3 of Paper I for a detailed description of the reference systems).

If we assume that the perturbation ${\boldsymbol{A}}$ is nearly constant (i.e., phase-independent) along the orbit, then under the adiabatic approximation, Equations (44) and (45) of Paper I orbit-averaged become

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\hat{{\boldsymbol{p}}},\hat{{\boldsymbol{q}}},\hat{{\boldsymbol{l}}}}^{\mathrm{ad}}=0\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\hat{{\boldsymbol{p}}},\hat{{\boldsymbol{q}}},\hat{{\boldsymbol{l}}}}^{\mathrm{ad}}=-\displaystyle \frac{3}{2}\displaystyle \frac{{(1+{e}^{2})}^{1/2}}{{na}}{A}_{q},\end{eqnarray} \tag{ 17 }$$

while Equations (46) and (47) of Paper I yield

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\hat{{\boldsymbol{r}}},\hat{{\boldsymbol{\tau }}},\hat{{\boldsymbol{n}}}}^{\mathrm{ad}}=0,\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\hat{{\boldsymbol{r}}},\hat{{\boldsymbol{\tau }}},\hat{{\boldsymbol{n}}}}^{\mathrm{ad}}=-\displaystyle \frac{3}{2}\displaystyle \frac{{(1+{e}^{2})}^{1/2}}{{na}}{A}_{q}.\end{eqnarray} \tag{ 19 }$$

Equations (16) and (17) describe the secular evolution of the osculating semimajor axis and eccentricity for a phase-independent perturbation under the adiabatic approximation and in the reference system ${K}_{J}(\hat{{\boldsymbol{p}}},\hat{{\boldsymbol{q}}},\hat{{\boldsymbol{l}}})$ shown in Figure 1. Equations (18) and (19) describe the same but in the reference system ${K}_{R}(\hat{{\boldsymbol{r}}},\hat{{\boldsymbol{\tau }}},\hat{{\boldsymbol{n}}})$ also shown in Figure 1.

In the case of isotropic wind mass loss, there is spherical symmetry, and thus the quantity ${\boldsymbol{b}}$ defined by Equation (3) is zero, i.e., ${\boldsymbol{b}}=0$. Since there is no mass transferred, we can set h = 0. We assume that the mass lost does not remove any extra angular momentum other than that lost due to the total mass change, i.e., $\zeta =0$. Substituting ${\boldsymbol{b}}=h=\zeta =0$ into Equations (1) and (2) and performing the averaging rule (14) under the adiabatic approximation ($\dot{M}/M$ constant along the orbit) leads to the secular time-evolution equations for the osculating semimajor axis and eccentricity in the adiabatic regime in the case of isotropic wind mass loss (e.g., Hadjidemetriou 1963):

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{iso}}^{\mathrm{ad}}=-\displaystyle \frac{\dot{M}}{M}a\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{iso}}^{\mathrm{ad}}=0,\end{eqnarray} \tag{ 21 }$$

where M is the total mass, $\dot{M}$ is the mass loss rate, and the subscript iso stands for isotropic.

Equations (20) and (21) indicate that an isotropic wind mass loss leads to an increase in the secular semimajor axis while keeping the secular eccentricity unchanged. In other words, under adiabatic isotropic wind mass loss, an initially circular orbit will remain circular on average while an initially eccentric orbit will retain on average its initial eccentricity.

We note here that Gurfil & Belyanin (2008) considered the case in which only one body is losing mass isotropically. Following Equation (3) in their paper, for this case they assume a perturbation of the form ${\boldsymbol{F}}=-({\dot{M}}_{1}/{M}_{1})\dot{{\boldsymbol{r}}}$ instead of ${\boldsymbol{F}}=-({\dot{M}}_{1}/M)\dot{{\boldsymbol{r}}}$. However, the latter is the correct perturbation for the system if only one component is losing mass in an isotropic manner (see Section 4 in Paper I). The former perturbation refers to the case of anisotropic ejection of mass with an absolute velocity of zero (see Section 6 here). This important difference leads to a factor of 2 discrepancy as well as a different denominator in their Equations (31), where $\dot{a}=-2({\dot{M}}_{1}/{M}_{1})a,\dot{e}=0$. The semimajor axis secular evolution equation is different compared to our Equation (20) due to both the different denominator and the factor of 2 discrepancy, while the eccentricity secular evolution remains zero in both.

In Paper I, we used the generalized Lagrange gauge ${\boldsymbol{\Phi }}=-\partial {\rm{\Delta }}L/\partial \dot{{\boldsymbol{r}}}$ (see Section 2.1.2 of Paper I) to derive the phase-dependent time-evolution equations for the contact orbital elements (see Equations (73) and (74) in Paper I). If we orbit-average these equations under the adiabatic approximation, we see that the secular evolution of these elements is zero, which is different from what Equations (20) and (21) indicate for the osculating semimajor axis and eccentricity. However, here we recall that unlike the osculating orbital elements, the contact orbital elements have no physical meaning but are sometimes useful mathematical tools (see Section 2.1.2 of Paper I for a more detailed discussion). In other words, this means that the fact that the contact orbital elements do not evolve secularly has no physical interpretation.

Under the assumption of isotropic delta-function mass loss at periastron, the orbit-averaging procedure (14) on Equations (1) and (2) (after the substitution ${\boldsymbol{b}}=h=\zeta =0$) yields

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{iso}}^{\delta }=-\displaystyle \frac{\dot{M}}{M}{(1-{e}^{2})}^{1/2}a\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{iso}}^{\delta }=-\displaystyle \frac{\dot{M}}{M}{(1-{e}^{2})}^{1/2}(1-e).\end{eqnarray} \tag{ 23 }$$

Equations (22) and (23) indicate that isotropic wind mass loss at periastron leads to a secular increases in the both the semimajor axis and eccentricity.

We note here that throughout the whole process, no assumption is made for low eccentricities, and so the above equations are valid for any value of eccentricity.

Besides, Equations (22) and (23) show that the circular case is equivalent to the adiabatic isotropic wind mass loss case (see Equations (20) and (21)). This can be seen by plugging e = 0 into Equations (22) and (23).

For completeness, we note that for a perturbing force linearly proportional to the orbital velocity, i.e., ${\boldsymbol{F}}\propto \dot{{\boldsymbol{r}}}$ (e.g., adiabatic isotropic wind mass loss studied here), the time-evolution of ω is $\propto \mathrm{sin}\nu$ which means that on average the change in ω is zero.

In Paper I, we derived the perturbing acceleration in the case of anisotropic wind mass loss from the donor of mass ${M}_{1}(t)$, which is given by ${{\boldsymbol{A}}}_{\mathrm{aniso}}={{\boldsymbol{F}}}_{\mathrm{aniso}}/{M}_{1}(t)$. For the sake of simplicity, we choose the reference system ${K}_{J}(\hat{{\boldsymbol{p}}},\hat{{\boldsymbol{q}}},\hat{{\boldsymbol{l}}})$ (see Figure 1). Substituting the aforementioned perturbing acceleration ${\boldsymbol{A}}$ into the adiabatic Equations (16) and (17) leads to the following secular time-evolution equations for the osculating semimajor axis and the eccentricity in the adiabatic regime for the case of anisotropic wind mass loss:

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\hat{{\boldsymbol{p}}},\hat{{\boldsymbol{q}}},\hat{{\boldsymbol{l}}}}^{\mathrm{ad},\mathrm{aniso}}=0\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\hat{{\boldsymbol{p}}},\hat{{\boldsymbol{q}}},\hat{{\boldsymbol{l}}}}^{\mathrm{ad},\mathrm{aniso}}=-\displaystyle \frac{3}{2}\displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{{naM}}_{1}(t)}{F}_{\mathrm{aniso},q}.\end{eqnarray} \tag{ 25 }$$

Here, the components of the perturbing force ${{\boldsymbol{F}}}_{\mathrm{aniso}}$ in the inertial reference system ${K}_{I}(\hat{{\boldsymbol{x}}},\hat{{\boldsymbol{y}}},\hat{{\boldsymbol{z}}})$ (see Figure 1) are given by

$$\begin{eqnarray}&&{F}_{\mathrm{aniso},x}=\displaystyle \frac{1}{4\pi }{\int }_{0}^{\pi }{\int }_{0}^{2\pi }J(\phi ,\theta ,t)u(\phi ,\theta ,t)d\phi \mathrm{sin}\theta \mathrm{cos}\phi d\theta \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{F}_{\mathrm{aniso},y}=\displaystyle \frac{1}{4\pi }{\int }_{0}^{\pi }{\int }_{0}^{2\pi }J(\phi ,\theta ,t)u(\phi ,\theta ,t)d\phi \mathrm{sin}\theta \mathrm{sin}\phi d\theta \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{F}_{\mathrm{aniso},z}=\displaystyle \frac{1}{4\pi }{\int }_{0}^{\pi }{\int }_{0}^{2\pi }J(\phi ,\theta ,t)u(\phi ,\theta ,t)d\phi \mathrm{cos}\theta d\theta .\end{eqnarray} \tag{ 28 }$$

Also, the transformation ${K}_{I}\to {K}_{J}$ is performed as follows using the matrix ${\boldsymbol{Q}}$ defined by Equations (52)–(60) in Paper I:

$$\begin{eqnarray}&&\left(\begin{array}{l}{F}_{\mathrm{aniso},p}\\ {F}_{\mathrm{aniso},q}\\ {F}_{\mathrm{aniso},l}\end{array}\right)=\left(\begin{array}{lll}{Q}_{11} & {Q}_{21} & {Q}_{31}\\ {Q}_{12} & {Q}_{22} & {Q}_{32}\\ {Q}_{13} & {Q}_{23} & {Q}_{33}\end{array}\right)\left(\begin{array}{l}{F}_{\mathrm{aniso},x}\\ {F}_{\mathrm{aniso},y}\\ {F}_{\mathrm{aniso},z}\end{array}\right).\end{eqnarray} \tag{ 29 }$$

It is obvious from Equation (24) that for a phase-independent perturbing force and under the adiabatic approximation, there is no secular evolution of the osculating semimajor axis due to phase-independent anisotropic wind mass loss (see also Veras et al. 2013). On the other hand, according to Equation (25), a secular change will be induced to the eccentricity if ${F}_{\mathrm{aniso},q}\ne 0$. The validity of this relation depends on the form and structure of the anisotropic wind. In the next two subsections, we briefly and qualitatively discuss the effect of different kinds of anisotropic wind structures (including jets) on the secular eccentricity.

5.1. Latitudinal and Longitudinal Dependence

5.2. Jets

6.1. Conservative Case $\dot{M}={\dot{J}}_{\mathrm{orb}}=0$

In all of the following sections, we choose to decompose our perturbations in the reference system K_R (see Figure 1).

When all of the mass ejected from the donor is accreted by the companion, there is no change in the total mass and orbital angular momentum of the system. For the so-called conservative case, this leads to $\dot{M}=0$ in Equations (1) and (2).

The secular time-evolution Equations (1) and (2) for the case of conservative mass transfer (i.e., $\dot{M}=0$) in the adiabatic regime for extended bodies lead to

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{con}}^{\mathrm{ad}} & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[e\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & +\langle {b}_{\tau }(1+e\mathrm{cos}\nu )\rangle ]-2{ha}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{con}}^{\mathrm{ad}} & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & \left.+\left\langle {b}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right\rangle \right],\end{eqnarray} \tag{ 31 }$$

while for a delta-function mass loss/transfer at periastron we have

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{con}}^{\delta } & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[e\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & +\langle {b}_{\tau }(1+e\mathrm{cos}\nu )\rangle ]-2{ha}{(1-{e}^{2})}^{1/2}\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{con}}^{\delta } & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & \left.+\left\langle {b}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right\rangle \right]\\ & & -2h{(1-{e}^{2})}^{1/2}(1-e).\end{eqnarray} \tag{ 33 }$$

We note here that the first terms in square brackets on the right-hand side (rhs) of Equations (30)–(33) depend through the quantity ${\boldsymbol{b}}$ defined in Equation (3) on the ejection and accretion points and associated velocities. The second terms on the rhs emerge from the mass transferred from the donor to the accretor and are proportional through the quantity h defined in Equation (4) to the mass loss/transfer rate ${\dot{M}}_{2}=-{\dot{M}}_{1}$.

In Paper I, we used the generalized Lagrange gauge ${\boldsymbol{\Phi }}=-\partial {\rm{\Delta }}L/\partial \dot{{\boldsymbol{r}}}$ (see Section 2.1.2 of Paper I) to derive the phase-dependent time-evolution equations for the contact elements (see Equations (105) and (106) in Paper I). Either under the adiabatic approximation or the delta-function mass loss/transfer at periastron, Equations (105) and (106) in Paper I for the secular contact semimajor axis and eccentricity give

$$\begin{eqnarray}{\langle \displaystyle \frac{{{da}}_{\mathrm{con}}}{{dt}}\rangle }_{\mathrm{con}}^{\mathrm{ad},\delta } & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[e\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & +\langle {b}_{\tau }(1+e\mathrm{cos}\nu )\rangle ]\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}{\langle \displaystyle \frac{{{de}}_{\mathrm{con}}}{{dt}}\rangle }_{\mathrm{con}}^{\mathrm{ad},\delta } & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & \left.+\left\langle {b}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right\rangle \right].\end{eqnarray} \tag{ 35 }$$

From Equations (35) and (31) we see that the secular evolution for contact and osculating eccentricity under the adiabatic approximation is the same, while from (34) and (30) we see that the secular evolution of the contact and osculating semimajor axis is different.

Under delta-function mass loss/transfer at periastron, Equations (34) and (32) as well as (35) and (33) show that both the secular evolution of contact and osculating eccentricity and semimajor axis are different. We again note that the contact orbital elements have no physical meaning but still remain useful mathematical tools.

6.1.1. Conservative Delta-function RLOF

In the case of RLOF, the mass is lost from the donor through the Lagrangian point L₁. This means that for RLOF conservative mass transfer, we have ${{\boldsymbol{r}}}_{{A}_{1}}={{\boldsymbol{r}}}_{{L}_{1},P},{{\boldsymbol{r}}}_{{A}_{2}}={{\boldsymbol{R}}}_{\mathrm{acc}}$, where ${L}_{1},P$ is the location of the Lagrangian point at periastron with respect to the center of mass of the donor and ${{\boldsymbol{R}}}_{\mathrm{acc}}$ is the radius vector of the accretor's surface with respect to its center of mass.

Sepinsky et al. (2007) assumed RLOF conservative delta-function mass loss/transfer at periastron so that ${\dot{M}}_{2}=-{\dot{M}}_{1}=-({\dot{M}}_{0}/2\pi )\delta (\nu )$. Although not explicitly stated but implied by their Equation (35), they assumed so-called stationary particles which have zero absolute velocities, i.e., (3) ${{\boldsymbol{w}}}_{1}={{\boldsymbol{w}}}_{2}=0$. Under this assumption, Equation (3) becomes

$$\begin{eqnarray}{\boldsymbol{b}} & = & {\dot{M}}_{2}\left(+\displaystyle \frac{{\omega }_{\mathrm{orb}}\times {{\boldsymbol{r}}}_{{A}_{2}}}{{M}_{2}}+\displaystyle \frac{{\omega }_{\mathrm{orb}}\times {{\boldsymbol{r}}}_{{A}_{1}}}{{M}_{1}}\right)\\ & & +{\ddot{M}}_{2}\left(\displaystyle \frac{{{\boldsymbol{r}}}_{{A}_{2}}}{{M}_{2}}+\displaystyle \frac{{{\boldsymbol{r}}}_{{A}_{1}}}{{M}_{1}}\right).\end{eqnarray} \tag{ 36 }$$

From Equations (32) and (33), we see that for delta-function mass loss/transfer, since $\mathrm{sin}\nu =0$ at periastron and both ${{\boldsymbol{r}}}_{{L}_{1},P}$ and ${{\boldsymbol{R}}}_{\mathrm{acc}}$ are not phase-dependent quantities, the only term that depends on ${\boldsymbol{b}}$ and survives in Equations (32) and (33) is the term including the tangential component of ${\boldsymbol{b}}$, i.e., ${b}_{\tau }$.

Substituting Equations (4) and (36) into (41) and (42) and after some algebra, we are led to the following secular time-evolution equations for the osculating semimajor axis and eccentricity in the case of conservative RLOF (Sepinsky et al. 2007):

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{con}}^{\delta ,\mathrm{RL}} & = & -2a\displaystyle \frac{{\dot{M}}_{2}}{{M}_{1}}\displaystyle \frac{1}{{(1-{e}^{2})}^{1/2}}\left[{qe}\displaystyle \frac{{r}_{{A}_{2}}}{a}\mathrm{cos}{\phi }_{p}+e\displaystyle \frac{{r}_{{A}_{1}}}{a}\right]\\ & & -2{\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right)a{(1-{e}^{2})}^{1/2}\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{con}}^{\delta ,\mathrm{RL}} & = & -\displaystyle \frac{{\dot{M}}_{2}}{{M}_{1}}{(1-{e}^{2})}^{1/2}\left[q\displaystyle \frac{{r}_{{A}_{2}}}{a}\mathrm{cos}{\phi }_{p}+\displaystyle \frac{{r}_{{A}_{1}}}{a}\right]\\ & & -2{\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right){(1-{e}^{2})}^{1/2}(1-e),\end{eqnarray} \tag{ 38 }$$

where $q={M}_{1}/{M}_{2}$ is the mass ratio and ${\phi }_{p}$ is the angle between the unit vector $\hat{{\boldsymbol{r}}}$ and the vector from the center of mass of the accretor to the accretion point A₂ at periastron.

Equations (37) and (38) are the same as Equations (39) and (40) of Sepinsky et al. (2007). Sepinsky et al. (2007) compared the timescale for the evolution of the eccentricity due to tides and RLOF and found that, in contrast to tides which always act to circularize the orbit, mass transfer may either increase or decrease the eccentricity over timescales ranging from a few Myr to a Hubble time. Furthermore, the timescale over which mass transfer acts to increase the eccentricity may be shorter than the tidal timescale which acts to decrease it.

For completeness, we note here that Equations (37) and (38) reduce to the point-mass for stationary particles Equations (70) and (71) described in Section 7.4 by setting ${{\boldsymbol{r}}}_{{A}_{1}}={{\boldsymbol{r}}}_{{A}_{2}}=0$. Thus, the first two terms in the rhs of Equations (37) and (38) emerge from the fact that we took into account the physical extent of the ejection and accretion point, which introduces additional perturbations to our problem. These perturbations can be seen by looking into terms dependent on ${{\boldsymbol{r}}}_{{A}_{1}},{{\boldsymbol{r}}}_{{A}_{2}}$ in Equation (3).

6.2. Non-conservative Case, $\dot{M},{\dot{J}}_{\mathrm{orb}}\ne 0$

When a fraction of the mass ejected from the donor is accreted by the companion and the residual mass is lost from the system, there is a change in the total mass and orbital angular momentum of the system. For this so-called non-conservative case, $\dot{M}\ne 0$ and in the most general case $\zeta \ne 0$ in Equations (1) and (2). From now on, in all cases with $\zeta \ne 0$ when we perform the orbit-average in the adiabatic regime, we assume that the parameter ζ is changing adiabatically so that we can take it out from the orbit-averaging integral (14).

The secular time-evolution Equations (1) and (2) for the case of non-conservative mass transfer (i.e., $\dot{M}=0$) in the adiabatic regime for extended bodies lead to

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{non} \mbox{-} \mathrm{con}}^{\mathrm{ad}} & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[e\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & +\langle {b}_{\tau }(1+e\mathrm{cos}\nu )\rangle ]-2{ha}\\ & & -(\zeta +1)\displaystyle \frac{\dot{M}}{M}\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{non} \mbox{-} \mathrm{con}}^{\mathrm{ad}} & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & \left.+\left\langle {b}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right\rangle \right],\end{eqnarray} \tag{ 40 }$$

while for delta-function mass transfer at periastron we have

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{non} \mbox{-} \mathrm{con}}^{\delta } & = & \displaystyle \frac{2}{n\sqrt{1-{e}^{2}}}[e\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & +\langle {b}_{\tau }(1+e\mathrm{cos}\nu )\rangle ]-2{ha}{(1-{e}^{2})}^{1/2}\\ & & -(\zeta +1)\displaystyle \frac{\dot{M}}{M}{(1-{e}^{2})}^{1/2}\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{non} \mbox{-} \mathrm{con}}^{\delta } & = & \displaystyle \frac{{(1-{e}^{2})}^{1/2}}{{na}}[\langle {b}_{r}\mathrm{sin}\nu \rangle \\ & & \left.+\left\langle {b}_{\tau }\displaystyle \frac{2\mathrm{cos}\nu +e(1+{\mathrm{cos}}^{2}\nu )}{1+e\mathrm{cos}\nu }\right\rangle \right]\\ & & -2h{(1-{e}^{2})}^{1/2}(1-e)\\ & & -(\zeta +1)\displaystyle \frac{\dot{M}}{M}{(1-{e}^{2})}^{1/2}(1-e).\end{eqnarray} \tag{ 42 }$$

Comparing Equations (30)–(33) with (39)–(42) we see an additional third term arising on the rhs of the above equations. This term emerges from the non-conservative nature of the mass transfer which carries away from the system both mass and orbital angular momentum. The first component of this term is proportional to the parameter ζ describing the loss in the system's orbital angular momentum, while the second component describes the change in the total system mass and is proportional to the total mass loss rate $\dot{M}$.

The additional effect induced by extra angular momentum removal from the system can be seen in Equations (39) and (40) or (41) and (42). Assuming an adiabatically changing $\zeta \ne 0$, this additional removal of angular momentum enhances the increase rate of the secular semimajor axis and has no effect on the secular eccentricity. Within the same considerations, the limiting case $\zeta \to 1$ does not induce any secular change in eccentricity but speeds up to the maximum level possible from this effect the secular expansion of the orbit. In the case of delta-function mass transfer at periastron, $\zeta \ne 0$ amplifies the increase rate of the semimajor axis and, depending on the system under study, speeds up the increase or decrease of eccentricity. Similar to the adiabatic case, the limiting case $\zeta \to 1$ maximizes the aforementioned effects.

6.2.1. Non-conservative Delta-function RLOF

For non-conservative delta-function RLOF, we assume that only a fraction γ of the lost mass is accreted such that ${\dot{M}}_{2}=-\gamma {\dot{M}}_{1}$. Following a procedure similar to that described in Section 6.1.1, Equations (1) and (2) provide the following for the secular time-evolution equations for the osculating semimajor axis and eccentricity in the case of non-conservative RLOF:

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{non} \mbox{-} \mathrm{con}}^{\delta ,\mathrm{RL}} & = & -2a\displaystyle \frac{{\dot{M}}_{2}}{{M}_{1}}\displaystyle \frac{1}{{(1-{e}^{2})}^{1/2}}\left[\gamma {qe}\displaystyle \frac{{r}_{{A}_{2}}}{a}\mathrm{cos}{\phi }_{p}\right.\\ & & \left.+\gamma e\displaystyle \frac{{r}_{{A}_{1}}}{a}\right]-2\displaystyle \frac{{\dot{M}}_{2}}{{M}_{1}}a{(1-{e}^{2})}^{1/2}[\gamma (q-1)\\ & & \left.+(1-\gamma )\left(\zeta +\displaystyle \frac{1}{2}\right)\displaystyle \frac{q}{1+q}\right]\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{non} \mbox{-} \mathrm{con}}^{\delta ,\mathrm{RL}} & = & -\displaystyle \frac{{\dot{M}}_{2}}{{M}_{1}}{(1-{e}^{2})}^{1/2}\left[\gamma q\displaystyle \frac{{r}_{{A}_{2}}}{a}\mathrm{cos}{\phi }_{p}\right.\\ & & \left.+\gamma \displaystyle \frac{{r}_{{A}_{1}}}{a}\right]-2\displaystyle \frac{{\dot{M}}_{2}}{{M}_{1}}{(1-{e}^{2})}^{1/2}(1-e)[\gamma (q-1)\\ & & \left.+(1-\gamma )\left(\zeta +\displaystyle \frac{1}{2}\right)\displaystyle \frac{q}{1+q}\right].\end{eqnarray} \tag{ 44 }$$

Equations (43) and (43) do not qualitatively change the results for the mass transfer timescale and the mass transfer effect on the secular semimajor axis and eccentricity we mentioned in Section 6.1.1 for the case of conservative mass transfer.

These equations are also derived in Sepinsky et al. (2009). However, in that study, it was assumed that mass is lost only from the accretor and not from the system as a whole as we assume here, and is described in detail in Section 6.2 in Paper I. This difference leads to the factor $\gamma (q-1)$ in our equations instead of $(\gamma q-1)$ in Equations (18) in (19) in Sepinsky et al. (2009). The same reason also leads to the missing γ factor in the first term on the rhs of Equation (18) and the second term on the rhs of Equation (19) in Sepinsky et al. (2009).

For the case where both binary components are treated as point-masses, we have derived the form of the perturbing force (see Equations (119)–(122) in Paper I) as well as the phase-dependent time-evolution Equations (8) and (9) for the semimajor axis and the eccentricity, respectively. In the following sections, we apply these equations to some commonly used characteristic examples of mass loss and mass transfer under the point-mass approximation. Furthermore, we compare our equations with equations in derived other studies that have considered applications similar to those presented here.

7.1. Radial Ejection from the Donor with No Accretion

We assume that the mass ejected from the donor is always in the radial direction pointing toward the accretor, while none of this matter is accreted. In this case, we have ${\dot{M}}_{2}=0,\dot{M}={\dot{M}}_{1},\zeta =0$ and ${{\boldsymbol{w}}}_{1}-{{\boldsymbol{v}}}_{1}={V}_{r}\hat{{\boldsymbol{r}}}$. The quantity ${\boldsymbol{c}}$ defined by Equation (10) then becomes

$$\begin{eqnarray}&&{{\boldsymbol{c}}}_{\mathrm{radial}}=-\displaystyle \frac{\dot{{M}_{1}}}{{M}_{1}}({{\boldsymbol{w}}}_{1}-{{\boldsymbol{v}}}_{1}),\end{eqnarray} \tag{ 45 }$$

from which in this case we have ${c}_{r}=-\tfrac{\dot{{M}_{1}}}{{M}_{1}}{V}_{r}$ and ${c}_{\tau }=0$ in Equations (8) and (9).

We introduce a parameter ${\lambda }_{r}$, which is a measure of the ejection velocity as a fraction of the orbital velocity at periastron and is defined by

$$\begin{eqnarray}&&{V}_{r}={\lambda }_{r}\displaystyle \frac{2\pi a}{P}.\end{eqnarray} \tag{ 46 }$$

The secular point-mass time-evolution Equations (8) and (9) for the semimajor axis and the eccentricity in the adiabatic regime for radial ejection with no accretion then become

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{r}^{\mathrm{ad}}=-\displaystyle \frac{{\dot{M}}_{1}}{M}a\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{r}^{\mathrm{ad}}=0.\end{eqnarray} \tag{ 48 }$$

From Equations (47) and (48), we see that a radial ejection from the donor with no accretion is equivalent to the case of isotropic wind mass loss, which was studied in Section 4 and is governed by Equations (20) and (21). Equations (47) and (48) are consistent with the work of Huang (1956).

7.2. Tangential Ejection from Donor with No Accretion

In this case, the mass is ejected in the tangential direction while the assumption of no accretion remains the same as in Section 7.1. This means that ${\dot{M}}_{2}=0,\dot{M}={\dot{M}}_{1},\zeta =0$ and ${{\boldsymbol{w}}}_{1}-{{\boldsymbol{v}}}_{1}=\pm {V}_{\tau }\hat{{\boldsymbol{\tau }}}$. In the last relation, for ${V}_{\tau }\gt 0$, the plus sign refers to ejection in the direction opposite to the direction of the motion of the body. We can then write the quantity ${\boldsymbol{c}}$ defined in Equation (10) as

$$\begin{eqnarray}&&{{\boldsymbol{c}}}_{\mathrm{tangential}}=-\displaystyle \frac{\dot{{M}_{1}}}{{M}_{1}}({{\boldsymbol{w}}}_{1}-{{\boldsymbol{v}}}_{1}),\end{eqnarray} \tag{ 49 }$$

from which, in this case, we have ${c}_{r}=0$ and ${c}_{\tau }=\mp \tfrac{\dot{{M}_{1}}}{{M}_{1}}{V}_{\tau }$ in Equations (8) and (9).

Here, we introduce the parameter ${\lambda }_{\tau }\gt 0$, which is defined in the same way as in Equation (46) by

$$\begin{eqnarray}&&{V}_{r}={\lambda }_{\tau }\displaystyle \frac{2\pi a}{P}.\end{eqnarray} \tag{ 50 }$$

The secular point-mass time-evolution Equations (8) and (9) for the semimajor axis and the eccentricity in the adiabatic regime for tangential ejection with no accretion then become

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{t}^{\mathrm{ad}}=-\displaystyle \frac{{\dot{M}}_{1}}{M}a\mp 2{\lambda }_{\tau }{(1-{e}^{2})}^{1/2}\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}a\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{t}^{\mathrm{ad}} & = & \mp {\lambda }_{\tau }\displaystyle \frac{{(1-{e}^{2})}^{3/2}}{e}\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}\\ & & \pm {\lambda }_{\tau }\left(\displaystyle \frac{{e}^{2}}{2}+1\right)\displaystyle \frac{{(1-{e}^{2})}^{1/2}}{e}\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}.\end{eqnarray} \tag{ 52 }$$

Here, we note that the plus sign refers to ejection in the direction of the motion of the body (front side) and the minus sign to the opposite direction (rear side). Equation (52) seems to have a singularity for circular orbits (e = 0). However, as we describe below, Equation (52) reduces to simpler forms with no singularity for either rear or frontal ejection.

Specifically, for rear ejection, tangential but opposite to the orbital motion, we have the minus sign and Equations (51) and (52) simplify to

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{t}^{\mathrm{ad}}=-\displaystyle \frac{{\dot{M}}_{1}}{M}a-2{\lambda }_{\tau }{(1-{e}^{2})}^{1/2}\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}a\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{t}^{\mathrm{ad}}=\displaystyle \frac{3}{2}{\lambda }_{\tau }e{(1-{e}^{2})}^{1/2}.\end{eqnarray} \tag{ 54 }$$

Equations (53) and (54) indicate that an ejection of mass opposite to the motion of the body direction leads to an increase in the secular semimajor axis and a decrease in the secular eccentricity of the orbit.

For frontal ejection, tangential to but along the orbital motion, we have the opposite result since in this case

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{t}^{\mathrm{ad}}=-\displaystyle \frac{{\dot{M}}_{1}}{M}a+2{\lambda }_{\tau }{(1-{e}^{2})}^{1/2}\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}a\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{t}^{\mathrm{ad}}=-\displaystyle \frac{3}{2}{\lambda }_{\tau }e{(1-{e}^{2})}^{1/2},\end{eqnarray} \tag{ 56 }$$

which shows that an ejection in the same direction as the motion of the body direction leads to a secular semimajor axis decrease and a secular eccentricity increase. Equations (53)–(56) for the circular case of e = 0 predict zero secular evolution for the eccentricity, which means that an initially circular orbit will remain circular under the assumption of only tangential adiabatic mass loss with no accretion in the case of point-masses. We note here that both quantitatively and qualitatively we agree with Huang (1956), who used a different formalism to derive equations similar to our Equations (53)–(56).

It is interesting to see at this point what happens in the rear ejection case for delta-function mass loss at periastron. In this case, we have

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{t}^{\delta }=-\displaystyle \frac{{\dot{M}}_{1}}{M}{(1-{e}^{2})}^{1/2}-\displaystyle \frac{{\lambda }_{\tau }}{\pi }(1-e)\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}a\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{t}^{\delta }=-\displaystyle \frac{{\lambda }_{\tau }}{\pi }{(1-e)}^{2}\displaystyle \frac{{\dot{M}}_{1}}{{M}_{1}}\end{eqnarray} \tag{ 58 }$$

which depicts that both the secular semimajor axis and eccentricity increase for rear delta-function mass loss at periastron. If the same ejection is to take place at apastron, then one can prove that this will lead to an increase in the secular semimajor axis and a decrease in the secular eccentricity.

Since the case of frontal ejection differs only by a sign compared to the rear ejection case, frontal delta-function mass loss at periastron will lead to a semimajor axis and eccentricity secular decrease, while delta-function mass loss at apastron will decrease the secular semimajor axis and increase the secular eccentricity.

7.3. Isotropic/Radial Wind Mass Loss from the Donor, Radial Partial Wind Accretion, and Tangential Mass Loss from the Accretor

In Section 7.1, we demonstrated that a radial ejection of mass in the adiabatic regime is secularly equivalent to the isotropic wind mass loss case studied in Section 4. Following the same logic, the same is true for any radially accreted matter.

For our case here, we assume that the donor is losing mass either isotropically in a wind or with a radial ejection always pointing toward the accretor. Part of this mass is accreted radially to the companion and the rest of it is tangentially lost from it with a constant absolute velocity ${\boldsymbol{V}}$. What introduces perturbing forces in the scenario described above is the tangential mass loss and we assume that is always occurs with a velocity opposite to the tangential motion of the accreting body. We note here that under these assumptions, the relative velocity of the tangentially lost mass to the center of mass of the companion is phase-dependent.

Based on the above assumptions, we have ${\dot{M}}_{2}=-\gamma {\dot{M}}_{1},\dot{M}=(1-\gamma ){\dot{M}}_{1}$ and ${{\boldsymbol{w}}}_{2}=V\hat{{\boldsymbol{\tau }}}$. Because the quantity ${\boldsymbol{c}}$ defined by Equation (10) is now phase-dependent, the tangential wind mass loss introduces an effective $\zeta ={\zeta }_{\mathrm{eff}}$. We note here that as we mentioned in Section 7.1, a radial ejection is in the adiabatic regime secularly equivalent to the isotropic wind mass loss and does not introduce any reaction force. Hence, the perturbing force defined by Equation (119) in Paper I is only due to the tangential mass loss and can be written as

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\mathrm{tan} \mbox{-} \mathrm{loss}}=\displaystyle \frac{\dot{{M}_{2}}}{{M}_{2}}({{\boldsymbol{w}}}_{2}-{{\boldsymbol{v}}}_{2})-\displaystyle \frac{1}{2}\displaystyle \frac{\dot{M}}{M}\dot{{\boldsymbol{r}}}\end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray}&&=\;\displaystyle \frac{\dot{{M}_{2}}}{{M}_{2}}V\hat{{\boldsymbol{\tau }}}-\left(\displaystyle \frac{1}{2}\displaystyle \frac{\dot{M}}{M}+\displaystyle \frac{\dot{{M}_{2}}}{{M}_{2}}\right)\dot{{\boldsymbol{r}}}.\end{eqnarray} \tag{ 60 }$$

From Equation (60), for the values of ${\boldsymbol{c}}$ and ${\zeta }_{\mathrm{eff}}$ we have

$$\begin{eqnarray}&&{\boldsymbol{c}}=\displaystyle \frac{\dot{{M}_{2}}}{{M}_{2}}V\hat{{\boldsymbol{\tau }}}\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{\zeta }_{\mathrm{eff}}=\displaystyle \frac{2\gamma }{\gamma -1}\left(\displaystyle \frac{{M}_{1}}{{M}_{2}}+1\right).\end{eqnarray} \tag{ 62 }$$

Based on Equations (61) and (62), in Equations (8) and (9) we determine that ${c}_{r}=0$, ${c}_{\tau }=\tfrac{\dot{{M}_{2}}}{{M}_{2}}V$, and $\zeta ={\zeta }_{\mathrm{eff}}$.

Similar to the previous subsections, here we introduce the parameter ${\lambda }_{a}$ such that

$$\begin{eqnarray}&&\displaystyle \frac{1-\gamma }{\gamma }V={\lambda }_{a}\displaystyle \frac{2\pi a}{P}.\end{eqnarray} \tag{ 63 }$$

We find that apart from the known secular equations for isotropic wind mass loss/radial ejection in the adiabatic regime

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }^{\mathrm{ad}}=-\displaystyle \frac{{\dot{M}}_{1}}{M}a\end{eqnarray} \tag{ 64 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }^{\mathrm{ad}}=0,\end{eqnarray} \tag{ 65 }$$

the secular time-evolution Equations (8) and (9) in the case studied here become

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }^{\mathrm{ad}}=-\displaystyle \frac{1-\gamma }{\gamma }\displaystyle \frac{{\dot{M}}_{2}}{M}a-2\lambda {(1-{e}^{2})}^{1/2}\displaystyle \frac{{\dot{M}}_{2}}{{M}_{2}}a\end{eqnarray} \tag{ 66 }$$

$$\begin{eqnarray}{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }^{\mathrm{ad}} & = & \left\{\displaystyle \frac{3}{2}\lambda e{(1-{e}^{2})}^{1/2}\right.\\ & & \left.+\rho \displaystyle \frac{{(1-{e}^{2})}^{3/2}}{e}\right\}\displaystyle \frac{{\dot{M}}_{2}}{{M}_{2}},\end{eqnarray} \tag{ 67 }$$

where $\rho ={(1-{e}^{2})}^{-1/2}-1$. In this case, Equations (66) and (67) indicate that the secular semimajor axis decreases while the secular eccentricity increases. We note here that Equations (64) and (65) are included in Equations (66) and (67).

Here, we note that we agree with the results by Boffin & Jorissen (1988). In that study, Boffin & Jorissen (1988), based on the work of Huang (1956), made some corrections and re-derived Equations (66) and (67). Bonačić Marinović et al. (2008) derived equations similar to our Equations (66) and (67) by making specific assumptions for the radial and tangential components of the wind velocity when the latter is accreted. As can be seen from the general Equations (1) and (2), the final expressions for the time-evolution of a and e depend strongly on the specific assumptions made about wind velocity when it is ejected or is accreted. Thus, comparison between the time-evolution equations for the semimajor axis and eccentricity between previous work is strongly driven by the different assumptions made separately in each of these works.

7.4. Conservative Mass Transfer for Stationary Particles

In many previous works, the ejected or accreted mass particles were assumed to not have an absolute velocity on their own, (e.g., Soker 2000; Eggleton 2006; Sepinsky et al. 2007). This assumption of the so-called stationary point particles leads to the additional relation ${{\boldsymbol{w}}}_{1}={{\boldsymbol{w}}}_{2}=0$, so that from Equation (3) we have ${\boldsymbol{b}}=0$ for this case.

Under these assumptions, Equations (1) and (2) lead to the following secular time-evolution equations for the semimajor axis and eccentricity in the adiabatic regime for stationary point particles and conservative mass transfer:

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{con}}^{\mathrm{ad}}=-2{\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right)a\end{eqnarray} \tag{ 68 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{con}}^{\mathrm{ad}}=0;\end{eqnarray} \tag{ 69 }$$

while for delta-function conservative mass transfer at periastron we have

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{con}}^{\delta }=-2{\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right)a{(1-{e}^{2})}^{1/2}\end{eqnarray} \tag{ 70 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{con}}^{\delta }=-2{\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right){(1-{e}^{2})}^{1/2}(1-e).\end{eqnarray} \tag{ 71 }$$

Equations (68)–(71) show that in the case of stationary particles, the effect of mass transfer on the secular semimajor axis and eccentricity depends on the mass ratio q. In the adiabatic regime, the secular eccentricity does not change, while the secular semimajor axis decreases for $q\gt 1$ and increases for $q\lt 1$. In the case of delta-function mass transfer at periastron, both the secular semimajor axis and the eccentricity decrease for $q\gt 1$ and increase for $q\lt 1$.

Here, we note that the same perturbation for conservative mass transfer and stationary particles, which is equal to $-{\dot{M}}_{2}\left(\tfrac{1}{{M}_{2}}-\tfrac{1}{{M}_{1}}\right)\dot{{\boldsymbol{r}}}$, is derived in Eggleton (2006; see our Equation (4) and Equation (6.27) on page 251 in Eggleton 2006).

However, for the case of delta-function mass transfer at periastron, Eggleton (2006) derives the following secular time-evolution equation for the eccentricity:

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{Egg}}^{\delta }=-2{\dot{M}}_{2}\left(\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}\right)(1+e),\end{eqnarray} \tag{ 72 }$$

which clearly has a very different functional dependence on e compared to our Equation (71). Formula (72) has also been used in previous work (e.g., Soker 2000). The reason for this discrepancy is that in these studies, there is an error in the computation of the orbit-averaging integral (14) for a delta-function conservative mass transfer in the case of stationary particles. In more detail, Soker (2000) made a substitution of $\mathrm{cos}\nu =1$ ($\nu =0$ at periastron) in Equation (2) instead of performing the complete averaging integral (14) with delta-function mass transfer at periastron.

Based on the Bondi–Hoyle accretion scenario (Bondi & Hoyle 1944), the accretion rate from a stellar wind can be written as

$$\begin{eqnarray}&&{\dot{M}}_{2}={a}_{\mathrm{BH}}\displaystyle \frac{4\pi {({{GM}}_{2})}^{2}}{{({V}_{\mathrm{rel}}^{2}+{c}_{w}^{2})}^{3/2}}{\rho }_{w},\end{eqnarray} \tag{ 73 }$$

where a_BH is the Bondi–Hoyle accretion efficiency, c_w is the wind sound speed, and ${{\boldsymbol{V}}}_{\mathrm{rel}}$ is the is the relative velocity of the wind with respect to the accreting star. An assumption commonly being made (e.g., Duncan & Lissauer 1998; Jura 2008; Villaver & Livio 2009; Passy et al. 2012; Spiegel & Madhusudhan 2012; Veras 2016) is that the density of the wind ${\rho }_{w}$ in the vicinity of the accretor follows the steady spherically symmetric profile ${\dot{M}}_{1}=-4\pi {r}^{2}{\rho }_{w}{V}_{w}$, where ${\dot{M}}_{1}$ is the wind mass loss rate and V_w is the wind velocity magnitude. Assuming that the wind flow is supersonic (${V}_{\mathrm{rel}}\gg {c}_{w}$), the accretion rate ${\dot{M}}_{2}$ for a radial wind (with respect to the orbit) of speed ${{\boldsymbol{V}}}_{w}={V}_{w}\hat{{\boldsymbol{r}}}$ is given by

$$\begin{eqnarray}&&{\dot{M}}_{2}=-{a}_{\mathrm{BH}}{\left(\displaystyle \frac{{{GM}}_{2}}{r}\right)}^{2}\displaystyle \frac{1}{{V}_{w}| {V}_{w}{\boldsymbol{r}}/r-\dot{{\boldsymbol{r}}}{| }^{3}}{\dot{M}}_{1}.\end{eqnarray} \tag{ 74 }$$

Here, the efficiency factor for Bondi–Hoyle accretion a_BH is in the range 0.5–1.0 in the Bondi–Hoyle model (e.g., Bondi & Hoyle 1944; Shima et al. 1985), although it may be as low as 0.025 in some specific cases (e.g., Theuns et al. 1996; Edgar 2004).

For fast wind, we have ${V}_{w}\gg {V}_{\mathrm{orb}}$ and Equation (74) reduces to

$$\begin{eqnarray}&&{\dot{M}}_{2}=| {\dot{M}}_{1}| \displaystyle \frac{{r}_{0}^{2}}{{r}^{2}},\end{eqnarray} \tag{ 75 }$$

where all of the constants have been summed up in r₀ to indicate dimensionality, and r₀ is defined by

$$\begin{eqnarray}&&{r}_{0}^{2}={a}_{\mathrm{BH}}\displaystyle \frac{{({{GM}}_{2})}^{2}}{{V}_{w}^{4}}.\end{eqnarray} \tag{ 76 }$$

For slow wind, we have ${V}_{w}\ll {V}_{\mathrm{orb}}$ and Equation (74) reduces to

$$\begin{eqnarray}&&{\dot{M}}_{2}=| {\dot{M}}_{1}| \displaystyle \frac{{r}_{0}^{2}}{{r}^{2}}{\left(\displaystyle \frac{{V}_{0}}{\dot{r}}\right)}^{3},\end{eqnarray} \tag{ 77 }$$

where again V₀ is a constant indicating dimensionality defined by ${V}_{0}={V}_{w}$.

Both cases are non-conservative mass transfer since only the fraction given by Equation (74) is accreted by the companion. This fraction is phase-dependent and, in the case of fast wind, is given by $\gamma (\nu )$, which is defined as ${\dot{M}}_{2}=-\gamma (\nu ){\dot{M}}_{1}$ and is given by

$$\begin{eqnarray}&&\gamma (\nu )=\displaystyle \frac{{r}_{0}^{2}}{{r}^{2}}=\displaystyle \frac{{r}_{0}^{2}{(1+e\mathrm{cos}\nu )}^{2}}{{a}^{2}{(1-{e}^{2})}^{2}}.\end{eqnarray} \tag{ 78 }$$

We then have $\dot{M}=(1-\gamma (\nu )){\dot{M}}_{1}$. Assuming point stationary masses (i.e., ${\boldsymbol{b}}=0$) and no extra angular momentum removal (i.e., $\zeta =0$), the total perturbing force for the phase-dependent partial fast wind accretion can be written as

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\mathrm{fw}}=-\left(\displaystyle \frac{{\dot{M}}_{2}}{{M}_{\mathrm{WT}}}+\displaystyle \frac{1}{2}\displaystyle \frac{\dot{M}}{M}\right)\dot{{\boldsymbol{r}}},\end{eqnarray} \tag{ 79 }$$

where the subscript fw stands for fast wind and for comparison we kept the same definition as in Eggleton (2006):

$$\begin{eqnarray}&&\displaystyle \frac{1}{{M}_{\mathrm{WT}}}=\displaystyle \frac{1}{{M}_{2}}-\displaystyle \frac{1}{{M}_{1}}.\end{eqnarray} \tag{ 80 }$$

Then, for the secular semimajor axis and eccentricity evolution due to fast wind Bondi–Hoyle (BH) partial accretion in the adiabatic regime, Equations (1) and (2) give the following equations:

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{da}}{{dt}}\right\rangle }_{\mathrm{fw}}^{\mathrm{BH}}=\displaystyle \frac{| {\dot{M}}_{1}| }{M}a-{\dot{M}}_{0}\displaystyle \frac{1+{e}^{2}}{1-{e}^{2}}a\left(\displaystyle \frac{2}{{M}_{\mathrm{WT}}}+\displaystyle \frac{1}{M}\right)\end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray}&&{\left\langle \displaystyle \frac{{de}}{{dt}}\right\rangle }_{\mathrm{fw}}^{\mathrm{BH}}=-{\dot{M}}_{0}e\left(\displaystyle \frac{2}{{M}_{\mathrm{WT}}}+\displaystyle \frac{1}{M}\right),\end{eqnarray} \tag{ 82 }$$

where for comparison with Eggleton (2006), we defined the quantity ${\dot{M}}_{0}$ as

$$\begin{eqnarray}&&{\dot{M}}_{0}=\displaystyle \frac{| {\dot{M}}_{1}| {r}_{0}^{2}}{{a}^{2}{(1-{e}^{2})}^{1/2}}.\end{eqnarray} \tag{ 83 }$$

We note that on page 252 and Equation (6.33) in Eggleton (2006), there is an error and the factor ${(1-{e}^{2})}^{2}$ in the denominator should be ${(1-{e}^{2})}^{1/2}$ as in our Equation (83).

After some algebra, we can see that the sign of the rhs of Equation (82) is determined by the sign of the quantity $2{q}^{2}+q-2$ with a single zero solution in the range $q\gt 0$ for q = 0.78. For $0\lt q\lt 0.78$, the rhs of Equations (81) and (82) is positive, and thus for a mass ratio q in the range $0\lt q\lt 0.78$ both the secular semimajor axis and eccentricity increase. For a mass ratio of $q\gt 0.78$, the rhs of Equation (82) is negative and the secular eccentricity always decreases, while the sign of the rhs of Equation (81) cannot be determined since it involves many and different parameters. This means that for a mass ratio $q\gt 0.78$, the semimajor secular evolution depends on many parameters including the orbital elements, the Bondi–Hoyle accretion efficiency parameter a_BH, the masses, and the fast wind absolute velocity V_W.

The general case of slow wind is described by Equation (77) and leads to an elliptical integral that demands numerical integration.

In the case that the wind velocity V_w is of the same order as the orbital velocity ${V}_{w}\sim {V}_{\mathrm{orb}}$, Equation (74) must be employed. This case requires numerical manipulation. The same is true for the any non-radial (with respect to the orbit) wind. However, assuming that the orbital velocity ${{\boldsymbol{V}}}_{\mathrm{orb}}=\dot{{\boldsymbol{r}}}$ is always perpendicular to the radial wind, which is always true in the circular case (Bondi–Hoyle circular approximation), then Equation (74) simplifies to

$$\begin{eqnarray}&&{\dot{M}}_{2}=-\displaystyle \frac{{a}_{\mathrm{BH}}}{2}{\left(\displaystyle \frac{{{GM}}_{2}}{{{rV}}_{w}^{2}}\right)}^{2}\left[\displaystyle \frac{1}{{(1+{V}^{2})}^{3/2}}\right]{\dot{M}}_{1},\end{eqnarray} \tag{ 84 }$$

where we have assumed a constant velocity wind, a Bondi–Hoyle factor a_BH, and we defined

$$\begin{eqnarray}&&{V}^{2}=\displaystyle \frac{{V}_{c}^{2}}{{V}_{w}^{2}}\end{eqnarray} \tag{ 85 }$$

$$\begin{eqnarray}&&{V}_{c}^{2}=\displaystyle \frac{G({M}_{1}+{M}_{2})}{a},\end{eqnarray} \tag{ 86 }$$

where the subscript c is a reminder of the circular approximation made here.

Under these approximations, Equation (84) reduces to an equation similar to the case of the fast wind Equation (75) but with r₀ defined by

$$\begin{eqnarray}&&{r}_{0}^{2}=\displaystyle \frac{{a}_{\mathrm{BH}}}{2}{\left(\displaystyle \frac{{{GM}}_{2}}{{V}_{w}^{2}}\right)}^{2}\left[\displaystyle \frac{1}{{(1+{V}^{2})}^{3/2}}\right].\end{eqnarray} \tag{ 87 }$$

Using r₀ defined in Equation (87), Equations (81)–(83) can be used accordingly to describe the Bondi–Hoyle accretion under the circular approximation.

Boffin & Jorissen (1988), Karakas et al. (2000), Hurley et al. (2002), and Vos et al. (2015) made use of Equation (84) but they first substituted $1/{r}^{2}$ by its average $\langle 1/{r}^{2}\rangle \;=1/[{a}^{2}{(1-{e}^{2})}^{1/2}]$. With this approach, they forced the mass transfer rate to be phase-independent, and so performing the averaging integral (14) in Equations (1) and (2) is oversimplified in their analyses. The right treatment involves orbit-averaging the full expression including the phase-dependent quantity $\gamma (\nu )$ given by Equation (78).

Moreover, in Equation (A.29) in Vos et al. (2015), the last terms on the rhs of Equations (1) and (2) are missing. These terms come from the change in total mass and orbital angular momentum of the system and cannot be neglected.

In this paper, we discussed the effect of various different types of mass loss and mass transfer processes on the orbital evolution of eccentric binary systems. We made use of the general formulation and the phase-dependent time-evolution equations of orbital elements presented in Paper I. Based on these equations, we derived the secular time-evolution equations for the semimajor axis and the eccentricity under either the adiabatic approximation or the assumption of delta-function mass loss/transfer at periastron.

We first discussed the cases of isotropic and anisotropic wind mass loss. We then studied non-isotropic ejection and accretion in a binary (including RLOF) in the conservative and non-conservative cases and for both point-masses and extended bodies. We presented specific examples of mass loss/transfer under the point-mass approximation and compared them with similar work in the literature. We concluded by discussing the case of phase-dependent mass accretion under the Bondi–Hoyle scenario.

The main results of the paper are summarized below.

• 1.  
  We studied the case of isotropic wind mass loss. We proved that under the adiabatic approximation, the secular osculating semimajor axis increases, while the secular osculating eccentricity remains constant. On the other hand, for delta-function isotropic mass loss at periastron, there is an increase in both the secular osculating semimajor axis and eccentricity.
• 2.  
  We studied the case of anisotropic wind mass loss. We showed that when the wind mass loss is phase-independent, the secular osculating semimajor axis is not affected. For wind mass loss that is longitudinally independent and symmetrical about the equator, the secular osculating eccentricity remains constant. However, any asymmetry about the equator or any existing longitudinal dependence introduces changes in the secular eccentricity that depend on the specific spatial structure of the wind in both latitude and longitude. Symmetric thin bipolar jets leave the secular eccentricity unchanged while any other asymmetrical or randomly directed jets induce secular eccentricity evolution.
• 3.  
  We continued with the non-isotropic ejection and accretion in the case of point-masses. In this case we find the following.

  • a.  
    A radial ejection from the donor with no accretion is secularly equivalent to the isotropic wind mass loss case mentioned above.
  • b.  
    An ejection of mass in the direction opposite to the motion of the body direction leads to an increase in the secular semimajor axis and a decrease in the secular eccentricity of the orbit. An ejection in the same direction as the motion of the body leads to a semimajor axis secular decrease and an eccentricity secular increase.
  • c.  
    For rear ejection and delta-function mass transfer at periastron, both the secular semimajor axis and eccentricity increase. In the case of apastron, this will lead to an increase in the secular semimajor axis and a decrease to the secular eccentricity. For frontal ejection, delta-function mass transfer at periastron will lead to a secular semimajor axis and eccentricity decrease, while delta-function mass transfer at apastron will decrease the secular semimajor axis and increase the secular eccentricity.
  • d.  
    In the case of isotropic/radial wind mass loss from the donor and radial partial wind accretion and tangential mass loss from the accretor, the secular semimajor axis decreases while the secular eccentricity increases.
• 4.  
  For the case of conservative or non-conservative RLOF, in contrast to tides which always act to circularize the orbit, mass transfer may either increase or decrease the secular eccentricity over timescales ranging from a few Myr to a Hubble time.
• 5.  
  For stationary particles, the secular semimajor axis and eccentricity time-evolution depends on the mass ratio $q={M}_{1}/{M}_{2}$. For the adiabatic mass loss rate, the secular eccentricity does not change while the secular semimajor axis decreases for $q\gt 1$ and increases for $q\lt 1$. In the case of delta-function mass loss at periastron, both the secular semimajor axis and eccentricity decrease for $q\gt 1$ and increase for $q\lt 1$.
• 6.  
  For phase-dependent mass transfer assuming point and stationary particles, applying the general Bondi–Hoyle accretion scenario requires numerical integration. However, the cases of fast wind or radial wind under the circular approximation are analytically integrable. For $0\lt q\lt 0.78$, both the secular semimajor axis and eccentricity increase. For a mass ratio $q\gt 0.78$, the secular eccentricity always decreases while the semimajor secular evolution depends on many parameters including the orbital elements, the Bondi–Hoyle accretion efficiency parameter a_BH, the masses, and the fast wind absolute velocity V_W.

The secular time-evolution equations for the semimajor axis and the eccentricity we presented here for various and different cases of mass loss/transfer could be used to model interacting eccentric binaries in star and binary evolution codes like StarTrack (Belczynski et al. 2008), BSE (Hurley et al. 2002), or MESA (Paxton et al. 2011, 2013, 2015).

In a future paper, we will attempt to relax the assumption of tidal locking in the case of extended bodies in the binary and study the effect of mass loss and mass transfer on the time-evolution of the binary orbital elements as well as the binary component spins. Assuming strong coupling between the spin and orbital angular momentum, we will derive secular time-evolution equations for the orbital elements and spins for the specific examples of mass loss and mass transfer studied in this paper, including new applications.

F.D. and V.K. acknowledge support from grant NSF AST-1517753 and Northwestern University through the "Reach for the Stars" program. The authors also thank our colleagues Dimitri Veras, Jeremy Sepinsky, Michael Efroimsky and Lorenzo Iorio who provided insight and expertise that greatly assisted the research. We are thankful to their comments that greatly improved the manuscript during its composition and revising stages.