We have presented the evolution of angular momentum and orbital period changes between the component spins and the orbit in close double white dwarf binaries undergoing mass transfer through direct impact accretion over a broad range of orbital parameter space. This work improves upon similar earlier studies in a number of ways: First, we calculate self-consistently the angular momentum of the orbit at all times. This includes gravitational, tides and mass transfer effects in the orbital evolution of the component structure models, and allow the Roche lobe radius of the donor star and the rotational angular velocities of both components to vary, and account for the exchange of angular momentum between the spins of the white dwarfs and the orbit. Second, we investigate the mass transfer by modeling the ballistic motion of a point mass ejected from the center of the donor star through the inner Lagrangian point. Finally, we ensure that the angular momentum is conserved, which requires the donor star spin to vary self-consistently. With these improvements, we calculate the angular momentum and orbital period changes of the orbit and each binary component across the entire parameter space of direct impact double white dwarf binary systems. We find a significant decrease in the amount of angular momentum removed from the orbit during mass transfer, as well as cases where this process increases the angular momentum and orbital period of the orbit at the expense of the spin angular momentum of the donor and accretor. We find that our analysis yields an increase in the predicted number of stable systems compared to that in the previous studies, survive the onset of mass transfer, even if this mass transfer is initially unstable. In addition, as a consequence of the tidal coupling, systems that come into contact near the mass transfer instability boundary undergo a phase of mass transfer with their orbital period.
Double white dwarf (DWD) binaries provide an interesting key to understand a variety of areas in astrophysics. Their birth properties provide insight into the evolution of their progenitors [
Following common envelope evolution, DWD binaries may emerge with a sufficiently short orbital period and rotational angular velocities, allowing gravitational radiation (GR), tides and mass transfer to drive the stars closer together on an astrophysically interesting time scale. Prior to the onset of mass transfer, their orbit will be shrink due to the effect of angular momentum via GR. Tidal forces are thought to synchronize the spin and orbital periods of the white dwarfs by the time that the white dwarfs are close enough together to begin transferring mass ( [
As the degenerate components of a DWD binary orbit continues to shrink via GR, the less massive component will inevitably fill its Roche lobe and begin transferring matter to its companion and the system will enter into a stable semi-detached state. In the case where the mass transfer is stable, such systems may be identifiable as AM CVn ( [
Mass transfer in close DWD binaries has generally been taken to be stable if the mass ratio is smaller than 0.8. However, [
As we have showed in [
The aim of this paper is to determine the evolution of angular momentum and orbital period changes between the component spins and the orbit in close DWD binaries throughout GR, tides and mass transfer effects by providing different range of orbital parameters and stellar models in a full self-consistent manner. In particular, we examine the comparison between numerical and analytical solutions to determine these systems whether, and under what circumstances, dissipative tidal coupling of the accretor to the orbit, through direct impact, can stabilize the dynamical mass transfer. This paper is organized as follows:
1 [
In Section (2) we develop the basic differential equations which governing the evolution of orbital parameters, assuming mass transfer through some well-specified model(s) and the basic assumptions with dynamical stability of mass transfer and discuss the difference between this method and the method utilized by the previous studies. In Section (3) we find and discuss the numerical solutions of the systems. In Section (4) we illustrate the results and discussion of this paper in comparison to the previous studies. Finally our main conclusion is summarized in Section (5).
In this paper, we consider a close binary system of two white dwarfs with masses Md and Ma, volume-equivalent radii of Ra and Rd, and uniform rotational angular velocities of Ω a and Ω d with axes perpendicular to the orbital plane for the accretor and donor, respectively2. We assume that the mass of each star is distributed spherically symmetrically. The binary is assumed to be in an initially circular keplerian orbit with orbital period, Porb. The radius of each object is assigned following Eggleton’s zero-temperature mass radius relationship of [
In this paper, we perform a Monte Carlo integration to calculate the volume-equivalent Roche lobe radius [
close DWD binaries, using the Eggleton function for the Roche lobe, R L , E g g for
calculating its size, the dynamical mass transfer rate, and the corresponding synchronization time scales for different cases, under certain restrictive approximations which determine the strength of tidal coupling. We now have all the steps in place to solve the evolution of these systems.
Our basic calculations are performed using a Monte Carlo integration method in a stationary inertial reference frame located at the initial center of mass of the binary system, which solving ordinary differential equations of orbital parameters at equivalent points. Following the above assumptions, we can investigate the outcomes for these systems with total mass, M and the orbital parameters such as Porb:
P o r b = ( 4 π 2 a 3 G M ) 1 / 2 , (1)
2Throughout this paper, the subscripts “a” and “d” will correspond to the accretor and donor, respectively.
3If the donor rotates non-synchronously, the Roche lobe radii can be calculated as given in [
where G and a are the universal gravitational constant and semi-major axis, respectively, and taking the donor to be the less massive star, the one that fills its Roche lobe.
As we have investigated in [
J t o t = J o r b + ∑ i = 1 2 ( J s p i n , i ) ; i ∈ ( a , d ) (2)
where J o r b is the orbital angular momentum, which is given by:
J o r b = M a M d ( G a M ) 1 / 2 , (3)
and the spin angular momenta of the accretor and donor are J s p i n , a = K a M a R a 2 Ω a and J s p i n , d = K d M d R d 2 ω d , respectively, where Ka and Kd are dimensionless constants depending upon the internal structure of the accretor and donor, respectively.
Here, we express the total angular momenta in terms of the rotational rates of the accretor, f a and donor, f d relative to the orbital angular velocities of the stars. Thus the rotational angular velocities of the stars, Ω i can be written in terms of the rotational rates and the angular velocity of the circular orbit, Ω o r b is given by
Ω i = f i Ω o r b . (4)
Then we can rewrite the total angular momentum of the binary system as:
J t o t = M a M d G a M + ∑ i = 1 2 ( K i M i R i 2 f i ) Ω o r b . (5)
The form of the J o r b term adopted above assumes the binary revolves at the Keplerian angular velocity of the circular orbit, Ω o r b 2 = G M / a 3 , which is a good approximation if the stars are centrally condensed.
Here, we introduce three effects that change the J o r b , P o r b and Ω i over time: mass transfer (MT), tides, and GR. Assuming each effect is independent of the others, we can then write the change of the orbital angular momentum as the sum of the change due to each of the above effects:
J ˙ o r b = J ˙ o r b , M T + J ˙ o r b , G R + ∑ i = 1 2 ( J ˙ o r b , t i d e s , i ) , (6)
which is known as the orbital angular momentum balance equation. To determine the total change in the angular momentum, then, we simply need to write the change due to each of the above effects. A system of two point masses orbiting around each other, in circular orbits, radiates gravitationally ( [
J ˙ o r b , G R = − ( 32 5 G 3 c 5 M a M d M a 4 ) J o r b . (7)
Prior to the onset of mass transfer, we assume that the spins and orbit to be synchronized, the binary orbit will shrink and the P o r b of the system will decrease due to the effect of angular momentum via GR as seen in Equation (7). During this time, tidal coupling will act to circularize the binary as well as synchronize the spins of the stars with the orbit. As mass transfer begins, angular momentum will be exchanged between the spin of the component stars and the orbit. This will ultimately affect the dynamical stability of the mass transfer process. As in [
J ˙ o r b , t i d e s = ∑ i = 1 2 ( K i M i R i 2 τ i ( f i − 1 ) Ω o r b ) ) . (8)
The term in the bracket of Equation (8) represent the torque due to dissipative coupling upon the accretor and donor. Hence, torques are parameterized in terms of the synchronization time scales ( [
In this work, we are only interested in direct impact mass transfer and orbital changes due to ballistic integrations where the particle impacts the surface of the accretor within one orbital period. In this case, the evolution of orbital parameters is determined by the differential equations, which will be given in Section (2.4). If the particle accretes within one orbital period, we classify this as a direct impact. If the particle does not accrete within one orbital period, it is likely that the ejection stream from the donor will self-intersect, resulting in the eventual formation of an accretion disk.
To determine J ˙ o r b , M T , we use the ballistic mass transfer calculations of [
J ˙ o r b , M T = ( Δ J o r b , b M P ) M ˙ d , (9)
where Δ J o r b , b is the change in the orbital angular momentum for a close DWD binaries as calculated by the above ballistic model for a single mass transfer event ejecting a particle of mass M P . In this method, changes in J o r b , M T are calculated at each time step by integrating the three-body system consisting of the two stars and the discrete particle representing the mass transfer stream. The change in the J o r b , M T per unit mass transferred is independent of the mass of the ejected particle as long as M P ≪ M d , M a .
As we investigated the rate of change of orbital angular momentum in Section (2.3), we develop the equations for the evolution of orbital period and the rotational angular velocities of the stars. It follows from Equation (3) and [
J ˙ o r b J o r b = ( 1 − q ) ( M ˙ d M d ) + 1 3 P ˙ o r b P o r b (10)
for conservative mass transfer ( M ˙ = 0 ) 4 and q = M d M a . Since M ˙ a = − M ˙ d .
Hence for the conditions M = constant and Jorb = constant it then follows that:
P ˙ o r b P o r b = − 3 ( M ˙ a M d ) ( q − 1 ) > 0. (11)
We examine the changes of the P o r b due to dynamical mass transfer, tides and GR. We assume that each effect is independent, and write the total change in the P o r b due to each of the above effects, respectively, as:
P ˙ o r b = P ˙ o r b , M T + P ˙ o r b , G R + ∑ i = 1 2 ( P ˙ o r b , t i d e s , i ) . (12)
To calculate P ˙ o r b , M T , we utilize the model for mass transfer developed in [
P ˙ o r b , M T = ( Δ P o r b , b M P ) M ˙ d . (13)
Here, Δ P o r b , b is the change in the orbital period for a close DWD binaries as calculated by the above ballistic model of [
4Recall that we assume the orbit remains circular in this paper.
Next we calculate P ˙ o r b , t i d e s and P ˙ o r b , G R . As noted by [
J ˙ o r b , t i d e s + J ˙ o r b , G R = ( 1 3 P o r b ( P ˙ o r b ) t i d e s , G R ) J o r b , (14)
where ( P ˙ o r b ) t i d e s , G R is the total change of the orbital period due to the combined
effects of tides and GR. Assuming as before that changes to the orbital period due to tides and changes due to GR are independent we can rewrite the above as:
J ˙ o r b , t i d e s + J ˙ o r b , G R = ( P ˙ o r b , t i d e s 3 P o r b + P ˙ o r b , G R 3 P o r b ) J o r b . (15)
Using Equation (3), Equation (7), and Equation (8), it follows that:
P ˙ o r b , t i d e s = 3 P o r b μ G a M ∑ i = 1 2 ( K i M i R i 2 τ i ω i ) , (16)
and
P ˙ o r b , G R = − 96 5 G 3 c 5 M a M d M a 3 . (17)
From Equation (4) and Equation (8) we can re-write Equation (16) as
P ˙ o r b , t i d e s = 3 μ ( P o r b M 2 ) 1 / 3 ( A a + D d ) , (18)
where
A a = K a M a R a 2 τ a ( Ω a − Ω o r b Ω o r b ) ,
and
D d = K d M d R d 2 τ d ( Ω d − Ω o r b Ω o r b ) . (19)
Finally, inserting Equation (13), Equation (17), & Equation (18) into Equation (12) leads to the equation for the dynamical evolution of the orbital period with time as:
P ˙ o r b = ( Δ P o r b , b M P ) M ˙ d + 3 ( A a + D d ) μ ( P o r b M 2 ) 1 / 3 − P ˙ o r b , G R . (20)
We derive the equations for the evolution of the components spins, Ω a and Ω d . Thus the changes in both components can be defined as:
Ω ˙ i = Ω ˙ i , M T + Ω ˙ i , t i d e s + Ω ˙ i , G R . (21)
Analogous to P ˙ M T in Equation (13), the change in Ω i due to mass transfer, Ω ˙ i , M T can be expressed as:
Ω ˙ i , M T = ( Δ Ω b , i M P ) M ˙ d . (22)
Here, Δ Ω b , i is the change in the rotational angular velocities of the star i resulting from a single mass transfer as described in Section (2.3). As in Equation (4), the spin angular momentum of each component can be written as:
J s p i n , i = K i M i R i 2 Ω i , (23)
where Ω i = f i Ω o r b . Hence, we consider that the tidal forces exist to redistribute angular momentum, working to keep the spins of the donor and accretor synchronous with the orbit ( f i = 1 ). Now we can determine Ω ˙ i , t i d e s and Ω ˙ i , G R by differentiating Equation (22) with respect to time assuming the mass held constant:
J ˙ s p i n , i = K i M i R i 2 ( Ω ˙ i , t i d e s + Ω ˙ i , G R )
− 2 3 K i M i R i 2 Ω i P o r b ( P ˙ t i d e s + P ˙ G R ) . (24)
The second term in Equation (24) depends only upon changes due to tides and GR. Since we do not include any GR effect on the spin angualar momentum of the components, conservation of angular momentum indicates that any changes in the spin angualar momentum of a component must be equal and opposite to the changes in the orbital angular momentum of the system due to tides acting on that component. Using Equation (4) and Equation (8), we have:
J ˙ s p i n , i = − ∑ i = 1 2 K i M i R i 2 τ i ( Ω i − Ω o r b ) . (25)
By combining Equation (24) and Equation (25) we can derive the expression for Ω ˙ i , t i d e s + Ω ˙ i , G R as:
Ω ˙ i , t i d e s + Ω ˙ i , G R = − Ω i τ i 2 3 Ω i P o r b [ − β G R + 3 ( A a + D d ) μ ( P o r b M 2 ) 1 / 3 ] , (26)
where
β G R = 96 G 3 M a M d M 5 c 5 a 3 . (27)
Using Equation (21), Equation (22) and Equation (26) we obtain the equations for the evolution of Ω a and Ω d as:
Ω ˙ a = ( Δ Ω b , a M P ) M ˙ d − Ω a τ a + 2 3 Ω a P o r b
[ − β G R + 3 ( A a + D d ) μ ( P o r b M 2 ) 1 / 3 ] , (28)
and
Ω ˙ d = ( Δ Ω b , d M P ) M ˙ d − Ω d τ d + 2 3 Ω d P o r b
[ − β G R + 3 ( A a + D d ) μ ( P o r b M 2 ) 1 / 3 ] , (29)
The nature of mass transfer and its stability is very important in a close DWD binaries. It is rooted in a similar treatment of mass transfer under consequential angular momentum losses via GR [
Δ = R d − R L , (30)
where R d is the radius of the donor and R L is the radius of its Roche lobe. The mass loss rate from the donor monotonically increases with Δ . The way in which the mass transfer rate varies with Δ has been investigated in many analyses (see, for example, [
M ˙ d = − f ( M a , M d , a , R d ) Δ 3 , (31)
for Δ > 0 , and zero for Δ < 0 . Combining results from [
f ( M a , M d , a , R d ) = 8 π 3 9 ( 5 G m e h 2 ) 3 / 2 ( μ ′ e m n ) 5 / 2 P o r b Λ , (32)
where Λ = ( 3 μ M d / 5 r L R d ) 3 / 2 [ a d ( a d − 1 ) ] − 1 / 2 , m e is the mass of an electron, m n is the mass of a nucleon, μ ′ e is the mean number of nucleons per free electron in the outer layers of the donor (which we will assume to be two) and μ and a d are parameters associated with the Roche potential
μ = M d M a + M d and a d = μ X L 1 3 + 1 − μ ( 1 − X L 1 ) 3 . (33)
where, X L 1 is the distance from the center of the donor to the inner Lagrangian point of the donor, in units of the semi-major axis, a.
In the case of [
r L = 0.49 q 2 / 3 0.69 q 2 / 3 + log ( 1 + q 1 / 3 ) , (34)
effectively, it is a tidal radius where mean density in lobes are equal:
R L = r L a , (35)
for 0.1 ≤ q ≤ 1 and so with the notation of [
R ˙ L R L = ζ r L M ˙ d M d + 2 3 P ˙ o r b P o r b , (36)
where ζ r L takes values between 0.33 and 0.48 is the logarithmic derivative of r L with respect to M d for both [
R ˙ d R d = D + ζ a d M ˙ d M d , (37)
where D = ( ∂ log ( R d ) / ∂ log ( ∂ t ) ) M d , represents the rate of change of the donor
radius due to intrinsic processes such as thermal relaxation and nuclear evolution, whereas ζ a d usually describes changes resulting from adiabatic variations of M d [
5The radius of each object is assigned following Eggleton’s zero temperature mass radius relation ( [
We integrate numerically the orbital evolution Equation (2), Equation (4), Equation (6), Equation (11), Equation (20), Equation (28), Equation (29), Equation (31), Equations (35)-(37) using 6th and 8th order Runge-Kutta ordinary differential equation solver [
evolution equations, we set ( ∂ log R d / ∂ t ) M d = 0 and use the zero-temperature mass-radius relationship.
In our integrations, we need to either assume or determine from other assumptions how the mass and angular momentum are redistributed in a close DWD binary system during mass transfer. This depends on the mode of accretion appropriate for the binary considered: does the stream impact the accretor or is an accretion disk present; is the mass transfer sub-Eddington and conservative, or are mass and angular momentum being ejected from the system following super-Eddington mass transfer. For most of the numerical integrations that follow, we use the appropriate rate of change of orbital angular momentum due to GR, MT and tides. However, if one is interested in the relatively rapid phases of dynamical mass transfer that follow contact and onset, then the qualitative properties of our integrated evolutions depend strongly on these assumptions.
As a system evolves, it is possible to pass back and forth through phases of mass transfer and phases of no mass transfer as the orbital period and the two masses change. If P o r b increases enough for mass transfer to stop altogether, the integration proceeds (with M ˙ d = 0 ) until the action of GR shrinks the orbit sufficiently for mass transfer to resume rather than the analytical one. We integrate over a period of 1.25 G yr. As in [
In
In this section, we investigate the numerical integrations for the change in the evolution of orbital angular momentum per unit transferred mass, which is independent of the mass of the ejected particle as long as M P ≪ M d from Equation (9) following a single ballistic orbit as a function of the donor mass for white dwarfs undergoing direct impact mass transfer.
the J ˙ o r b , M T and M ˙ d due to direct impact accretion with the P ˙ o r b and M a . As expected, our numerical results agree quite well to this analytic solution.
In
Following the above consideration, we show the orbital angular momentum changes per unit transferred mass in the parameter space of close DWD binaries with a low donor mass are more likely to remain stable over long periods of time, while the majority of the parameter space is expected to be unstable. Thus, we further investigate these parameter space from deformation to direct impact where the orbital angular momentum decreases more rapidly than previous studies, which shown as in solid red line.
It is commonly assumed ( [
In
As we developed and discussed the basic equations for the evolution of orbital
parameters in Sections (2) and (3), respectively, we compute the evolution over a grid in M d , M a and P o r b parameter space for two different tidal synchronization time scales at contact: τ = 10 15 years and τ = 10 years to determine the dynamical stability of various systems. We also elaborate the comparisons between numerical and analytical solutions for the evolution of angular momentum and orbital period changes due to direct impact accretion in close DWD binaries by various stellar model approximations of mass transfer, with particular attention payed to the dynamical stability of these processes against runaway on synchronization time scales of the mass donating star.
As we discussed in Section (3),
As noted by [
Stronger tidal coupling will allow more spin angular momentum from the accretor to be transferred back into the orbit, which correspondingly slower increase in orbital period, causing the mass transfer rate and the rate of change of orbital angular momentum to decrease. This is to be expected and in agreement with the analyses of [
In
to increase, even as the Roche lobe grows due to the increasing orbital period (see, e.g., Equation (30) and Equation (31)). Thus, we present the results of evolution for the orbital parameters of the tides generating body with tidal synchronization time scale of τ = 10 15 years . Here, we acknowledge the main limitation of our result to assume that systems are stable once they reach a phase of disc accretion. Systems with the less masses of the donor stars parameter space of interest here begin the integration in a disc phase, and are therefore labeled as stable. However, it is feasible that if these high donor mass systems were allowed to evolve through the initial disc phase, they would eventually become super-Eddington and potentially dynamically unstable.
We have computed different orbital parameters of close DWD binaries and their evolution with synchronization time scales using R L , E g g . Hence, for τ = 10 15 years , the system is dynamically unstable and for τ = 10 years the system is stable, but passes through a phase of super-Eddington accretion. In this section, we compare the analytical solutions of the various orbital parameters of these systems with the numerical solution that we investigated in Sections (3) and (4).
The rotational angular velocities of the component white dwarfs remain closer to synchronize. In our work, when we apply the R L , E g g , the mass transfer rate will continue to increase until tides have transferred sufficient angular momentum from the spin of the components to the orbit to expanded the orbit and decrease the mass transfer rate. Thus, the mass transfer rate is decreased and tides continue to redistribute angular momentum between the component spins and the orbit (see,
In
analytically investigate the evolution of the rotational angular velocities of both donor and accretor stars in close DWD systems with initial masses of the donor, M d = 0.86 M ⊙ and accretor, M a = 1.2 M ⊙ . In these systems, we provide a comparison between the numerical and analytical solution for the rate of change of the spin angular momentum for the accretor and the rate of change of the mass transfer rate corresponds to the Ω ˙ d and Ω ˙ a , which shows our calculation stops when the donor reached 1.45 M ⊙ (disc accretion).
As it was studied by [
As seen in Figures 1-5, we discussed the evolution of angular momentum and orbital changes due to direct impact and evolution of systems with strong tidal coupling. Also, in Figures 7-9, we investigated the comparison between the numerical and analytical solution of the systems for the evolution of parameters for τ = 10 years . However, as seen in
We have presented the evolution of angular momentum and orbital period changes between the component spins and the orbit in close DWD binaries undergoing mass transfer through direct impact accretion, including the effects due to mass transfer, gravitational radiation, and tidal forces between the donor, accretor, and the orbit. The theoretical framework that we have outlined in this paper can be used to generate the models for the rotational angular velocities and the orbital periods of close DWD binaries in general. Thus, we implemented the ballistic mass transfer treatment developed in [
A direct comparison of the results obtained from the numerical solutions in Section (3) with those of analytical solutions in Section (4) shows that both the numerical solution and the Eggleton approximation for the Roche lobe calculations are quite accurate.
In many cases the orbital angular momentum lost from the orbit can be significantly less than the standard assumption, making this process less destabilizing than expected. This may allow for more DWD to survive the dynamical instability of mass transfer rate and evolve into systems like AM CVn, instead of merging to create Type Ia supernovae evolving to higher separations and diminishing mass transfer rates. This result, which was not predicted by the analytical solution in [
In a few cases, we have shown that mass transfer may increase the orbital angular momentum of the orbit, thereby providing a stabilizing effect on the orbit. Hence, any stabilizing effect increases the chances of a long-lived close DWD binaries, lending acceptance to the creation of AM CVn through the DWD models.
We also account for the modification of the Roche lobe size due to the orbital period and ζ a d of the donor stars. As a result, we found that the number of stable systems increases, particularly for the case of strong tidal coupling, as shown in
Finally, the large scale numerical computations presented in this paper have provided the realistic models that describing the evolution of angular momentum and orbital period changes and its stability of mass transfer in close DWD binary systems.
We thank Ethiopian Space Science and Technology Institute (ESSTI)-Entoto Observatory and Research Center (EORC), Astronomy and Astrophysics Research Development Department for giving research opportunity and facilities. Negu S. H. thanks Jigjiga University for giving study leave. This research has made use of NASA’s Astrophysical Data System.
The authors declare no conflicts of interest regarding the publication of this paper.
Negu, S.H. and Tessema, S.B. (2018) Evolution of Angular Momentum and Orbital Period Changes in Close Double White Dwarf Binaries. International Journal of Astronomy and Astrophysics, 8, 275-298. https://doi.org/10.4236/ijaa.2018.83020