A comment on the properties of the matter flow through the first Lagrangian point

Ziółkowski, Janusz; Zdziarski, Andrzej A

doi:10.1093/mnras/stab2826

ABSTRACT

We analyse properties of the mass outflow from the Roche lobe filling component of a semidetached binary system. We follow the approaches published by Paczyński & Sienkiewicz and by Lubow & Shu, which we compare with other simplified approaches. We find that the density of the flow at L₁ is orders of magnitude lower than the density on the same equipotential but away from L₁. Furthermore, the effective cross-section of the flow, after averaging over its profile of the momentum density, is much lower than some published estimates done without accounting for the averaging. Thus, the use of some simplified formulae for the density and the flow cross-section can lead to overestimates of the accretion rate and of the mass contained in the L₁ regions by very large factors unless they are supported by simultaneous integrations of the equations of stellar structure for the outer layers of the donor.

binaries: mass transfer, stars: binary, stars: evolution, stars: general

1 INTRODUCTION

The knowledge of the mass transfer rate from the donor to accretor, |$\dot{M}$|⁠, in accreting stellar systems is of major importance for their understanding. By comparing the transfer rate with the accretion rate, we can assess whether the mass transfer is conservative or associated with mass-loss, as well as examine an effect of irradiation on the mass flow. The transfer rate then determines the past and future evolution of accreting systems. Observationally, |$\beta \dot{M}$| can be estimated from a long-term average of the luminosity in an X-ray binary, e.g. Coriat, Fender & Dubus (2012) and Zdziarski, Ziółkowski & Mikołajewska (2019), where β ≤ 1 is the fraction of the transferred mass accreted. However, for understanding of many aspects of these systems, it is important to estimate |$\dot{M}$| theoretically, and compare the two estimates. In semidetached binaries, such estimates can be done based on considering the evolutionary changes of the stellar radius compared to the changes of the Roche lobe, e.g. Webbink, Rappaport & Savonije (1983). If the mass transfer is driven by nuclear expansion of the donor, |$\dot{M}$| is connected to the time-scale of this expansion, see equation (4.16) of Frank, King & Raine (2002). Using both approaches, we can relatively reliably determine the rate of the mass transfer, |$\dot{M}$|⁠.

On the other hand, sometimes it is useful to consider the dynamics of the flow through the vicinity of the inner Lagrangian point of the Roche lobe, L₁. This is the case if we need to consider the conditions near the L₁ point, e.g. in order to investigate the effects of irradiation. Such analysis is helped by approximate formulae relating the radius excess, ΔR, i.e. the difference between the stellar radius and the radius of the Roche lobe, to the rate of the mass transfer |$\dot{M}$| (Paczyński & Sienkiewicz 1972, hereafter PS72, who used in part unpublished results of Jȩdrzejec 1969; Savonije 1978, 1979, 1983). On the other hand, Lubow & Shu (1975), hereafter LS75, related the physical conditions at L₁ to the sound speed and the binary parameters only. Their analysis has been widely applied in several contexts, see, e.g. Lai, Helling & van den Heuvel (2010).

The accuracy of these formulae remains an open question. Does the rate of the mass transfer can be obtained from either the radius excess or the sound speed? We would like to stress that the mechanism responsible for the determination of |$\dot{M}$| is the internal evolutionary engine of the donor. The structure of the outer stellar layers (including the vicinity of L₁) just adjusts to carry out whatever |$\dot{M}$| is set by the evolutionary engine. This includes both the radius excess and the gas density at L₁. Here, we test the accuracy of the formulae of PS72 and LS75, and find their applicability is not universal.

2 ESTIMATES OF THE FLOW PARAMETERS

2.1 The approach of PS72

Here, we recall and extend some of the results of PS72 and compare them to other expressions in literature. We use the coordinate system in the corotating frame as shown by PS72 in their fig. A1. The x-axis connects the stellar centres, and it is given in units of the semimajor axis (equal to the separation between the stars, A, for a circular orbit). The orbital motion is neglected, and thus the system has an axial symmetry around x, and the y-axis is perpendicular to x in any direction. We will also neglect a small difference between the cross-sections at x = 0 of the star overflowing its Roche lobe, y_f in the notation of PS72, and of the equipotential surface which the star fills, y_s. This results in a factor of slightly less than unity by which the true |$\dot{M}$| should be multiplied (see table A1 in PS72).

The dimensionless radius of the flow cross-section at x = 0 is then given by (using equation A15 of PS72 at y = 0):

$$\begin{eqnarray} y_{\rm s}\approx \left(\frac{2 A}{R} \frac{\Delta R}{R}\right)^{1/2} \frac{\mu ^{3/4}(1-\mu)^{1/4}}{1+2\sqrt{\mu (1-\mu)}}, \end{eqnarray}$$

(1)

where R is the donor radius, ΔR is the radius excess, i.e. the difference between the stellar radius and the radius of the Roche lobe, μ ≡ M/(M + M_X), M and M_X are the donor and accretor masses, respectively, A is given by the Kepler law,

$$\begin{eqnarray} A^3=\frac{P^2 G M}{4\pi ^2 \mu }, \end{eqnarray}$$

(2)

and P is the binary period. Equation (1) is based on purely geometrical considerations and as such is not subject to serious errors (other than neglecting the orbital motion and approximating y_f by y_s). The total cross-section of the flow at L₁ is given by |$\Sigma =\pi (A y_{\rm s})^2$|⁠.

Then, PS72 assume that pressure, p, is given by the polytropic relation,

$$\begin{eqnarray} p= K\rho ^{1+1/n}, \end{eqnarray}$$

(3)

where K is a constant. With that assumption, they find the distribution of the momentum density, ρv, as a function of the distance from the L₁ point along the y axis,

$$\begin{eqnarray} \rho v \approx \left\lbrace \begin{array}{@{}l@{\quad }l@{}}C \left(y_{\rm s}^2-y^2 \right)^{n+1/2}, &y\le y_s;\cr 0, &y\gt y_s,\cr \end{array}\right. \end{eqnarray}$$

(4)

where

$$\begin{eqnarray} C=\left(\frac{-\Omega _0 h}{n+1/2}\right)^{n+1/2}\left[K(1+1/n)\right]^{-n}, \end{eqnarray}$$

(5)

$$\begin{eqnarray} -\Omega _{0}= \frac{G M}{A\mu } \left[1+2\sqrt{\mu (1-\mu)}\right] \end{eqnarray}$$

(6)

$$\begin{eqnarray} h=\frac{1+2\sqrt{\mu (1-\mu)}}{2\sqrt{\mu (1-\mu)}}. \end{eqnarray}$$

(7)

We note here that ρv decreases quickly with the distance along y away from L₁, and its average value is given by

$$\begin{eqnarray} \langle \rho v\rangle = \frac{2}{2 n+3}\rho _{\rm \mathit{ L}_1} v_{\rm \mathit{ L}_1}, \end{eqnarray}$$

(8)

$$\begin{eqnarray} \rho _{\rm \mathit{ L}_1} v_{\rm \mathit{ L}_1}=\left[\frac{\Delta R}{R} \frac{G M}{R(1/2+ n)}\right]^{1/2+n}\left[ K(1+1/n)\right]^{-n}, \end{eqnarray}$$

(9)

where the subscript L₁ denotes a quantity measured at L₁.

We can also write in general

$$\begin{eqnarray} \dot{M}=\Sigma _{\rm eff}\rho _{\rm \mathit{ L}_1} v_{\rm \mathit{ L}_1}, \end{eqnarray}$$

(10)

where Σ_eff is the effective cross-section taking into account the averaging of equation (8),

$$\begin{eqnarray} \Sigma _{\rm eff}=\frac{\Delta R}{R}\frac{G M}{R} \frac{P^2\sqrt{\mu (1-\mu)}}{\pi (2n+3) \left[1+2\sqrt{\mu (1-\mu)}\right]^2}. \end{eqnarray}$$

(11)

If M ≲ 0.6M_X, we can use the approximation to the donor’s Roche lobe radius by Paczyński (1967), |$R\approx (2 G M)^{1/3} (P/9\pi)^{2/3}$|⁠, which yields

$$\begin{eqnarray} \Sigma _{\rm eff}=R\Delta R \frac{3^4\pi \sqrt{\mu (1-\mu)}}{2 (2n+3) \left[1+2\sqrt{\mu (1-\mu)}\right]^2}. \end{eqnarray}$$

(12)

Thus Σ_eff ∼ RΔR (as noted by Savonije 1979), which is much larger than a very simple, but incorrect, estimate of ∼(ΔR)². The Σ_eff of equation (11) can be compared with an estimate of Σ in Zdziarski, Wen & Gierliński (2007), which is an approximation to a formula of Savonije (1983),

$$\begin{eqnarray} \Sigma _0\approx \frac{\Delta R}{R}\frac{G M}{R} \frac{P^2}{2\pi }. \end{eqnarray}$$

(13)

This does not include the averaging and mass ratio terms, and is usually an overestimate.

Using |$\dot{M}=\Sigma _{\rm eff}\rho _{\rm \mathit{ L}_1} v_{\rm \mathit{ L}_1}$| together with equations (9) and (11), we obtain

$$\begin{eqnarray} \dot{M}\approx \left(\frac{\Delta R}{R} \frac{G M}{R}\right)^{n+3/2} \frac{P^2}{ K^n}\frac{\sqrt{\mu (1-\mu)}}{\left[1+2\sqrt{\mu (1-\mu)}\right]^2} C_n, \end{eqnarray}$$

(14)

$$\begin{eqnarray} C_n=\frac{1}{\pi (2n+3)(n+1/2)^{n+1/2}(1+1/n)^n}, \end{eqnarray}$$

(15)

which is in complete agreement with equations (A21), (A22), and (A17) of PS72 (who expressed |$\dot{M}$| in terms of A instead of P). Note that the first and second terms in equation (14) are given by very large and very small numbers, respectively, especially for a high n, and a care is needed in their numerical calculation.

We note that Savonije (1978) repeated the analysis of PS72 in a more precise way. In particular, he took into account the orbital motion of the matter in the binary system. His formulae are more complicated, but the general results are similar to those of PS72. In particular, he obtained the same functional form of the relations given here by equations (4) and (14); specifically, with the same power exponents.

Equation (14) can give us a reasonably accurate estimate of ΔR if we know |$\dot{M}$|⁠. This formula takes into account the physical state of the outflowing matter through the polytrope constant, K, and the polytropic exponent, n. Thus, in order to determine the values of these parameters we need a model of the outer layers of the donor. Importantly, we definitely need these values at the L₁ point rather than at the depth ΔR below the surface of the star far from L₁ (as practiced by some simplified formulae users). Also, if we use directly equation (10) with estimates of Σ, we need the values of density and pressure at L₁ rather than away from it at the depth ΔR.

Furthermore, we need an estimate of the velocity of the flow. The most natural assumption appears that the velocity of the matter passing through the x = 0 plane is similar to the sonic velocity, c_s. Arguments in favour of that were given by LS75 and Savonije (1978). If we assume that the matter crossing the x = 0 plane has at each point the velocity equal to the speed of sound, then for a polytrope

$$\begin{eqnarray} v^2 = c_{\rm s}^2=(1+1/n) K \rho ^{1/n}. \end{eqnarray}$$

(16)

We can then obtain the density profile at x = 0 with equation (4) to be given by

$$\begin{eqnarray} \rho (y)=\frac{C^{\frac{2n}{2n+1}} \left(y_{\rm s}^2-y^2\right)^n }{\left[ K(1+1/n)\right]^{\frac{n}{2n+1}} },\quad y\le y_{\rm s}, \end{eqnarray}$$

(17)

and |$\rho _{\rm \mathit{ L}_1}=\rho (0)$|⁠.

2.2 The approach of LS75

LS75 used a different approach than PS72. Their description of the flow through L₁ point is more physical and covers much larger range of effects than that of PS72. On the other hand, their formulae are less quantitative. They show that the dimensional flow radius is

$$\begin{eqnarray} r_{\rm f} \approx c_{\rm s}/\Omega, \end{eqnarray}$$

(18)

where c_s is the gas sound speed at L₁ and |$\Omega =2\pi /P$| is the binary orbital frequency. By comparing with equation (1), we see that the two become similar (apart from the μ-dependent term of the order of unity) if ΔR ∼ H_*, where H_* is the stellar scale height (since it is defined by |$c_{\rm s}^2=(H_*/R)(G M/R)$|⁠. We note that this fixes ΔR independently of the rate of the mass outflow and of the physical state of the outflowing matter (with the exception of its temperature), which does not seem to be generally valid. Still, at least in some cases (Section 3), it gives results that are surprisingly close to the results obtained using equation (9) and assuming sonic velocity of gas at L₁.

As a consequence of equation (18), LS75 further argue that

$$\begin{eqnarray} \rho _{\rm \mathit{ L}_1}\approx \dot{M} \frac{4\pi ^2}{P^2 c_{\rm s}^3}. \end{eqnarray}$$

(19)

The above equation is presented in the discussion of their equation (11). Using equation (10) with |$v_{\rm \mathit{ L}_1}=c_{\rm s}$|⁠, we have

$$\begin{eqnarray} \Sigma _{\rm eff} \approx r_{\rm f}^2= c_{\rm s}^2 \frac{P^2}{4\pi ^2}. \end{eqnarray}$$

(20)

We note that the same formulae (with the accuracy to a factor of |$\pi$|⁠) are used by Frank et al. (2002) (p. 352).

Also, LS75 estimated the ratio of the density at the Roche equipotential inside the star far from L₁, which we will denote as ρ₀, to that at L₁, as

$$\begin{eqnarray} \frac{\rho _0}{\rho _{\rm \mathit{ L}_1}} \sim \frac{\Omega A}{c_{\rm s}} \end{eqnarray}$$

(21)

(see discussion after their equation 64).

3 COMPARISON WITH A MODEL OF GX 339–4

As an example, we consider a model of the binary system GX 339–4. This system accretes from an evolved low-mass star on to a black hole, and its binary period is P = 1.7587 d. Specifically, we consider a model slightly modified with respect to the evolutionary model D obtained by Zdziarski et al. (2019), and assume |$\dot{M} \approx 6.1 \times 10^{16}$| g s⁻¹. This |$\dot{M}$| is equal to the value obtained from the evolutionary model by matching the rate of Roche lobe expansion to the stellar expansion. In that model, we have M = 1 M_⊙, R = 2.83 R_⊙, the accretor mass of M_X = 8 M_⊙, giving μ = 1/9, and the radius excess of ΔR ≈ 1.22 × 10⁹ cm (a slightly corrected value, following from equation 14). The stellar density at the depth ΔR at the equipotential describing the Roche lobe away from L₁ was found to be ρ₀ = 3.1 × 10⁻⁶ g cm⁻³ and the temperature was T₀ = 1.56 × 10⁴ K. The local value of the polytropic exponent, n, estimated at that depth, is n = 6.43. The matter there is almost fully ionized, and X = 0.74 and Z = 0.014 was assumed, which implies the mean molecular weight of 0.598. Using equation (3) and assuming ideal gas, we obtain the entropy parameter of K ≈ 1.55 × 10¹³ (cgs), and the sound speed of c_s0 ≈ 1.6 × 10⁶ cm s⁻¹. These values of K and n are then assumed to apply to the flow through L₁ and be constant within it.

For the above parameters, we can find the velocity and density at L₁ from equations (16) and (17), respectively. They are |$v_{\rm \mathit{ L}_1}\approx 7.8\times 10^5$| cm s⁻¹ and |$\rho _{\rm \mathit{ L}_1}\approx 3.4\times 10^{-10}$| g cm⁻³. Thus, the density at L₁ is as much as ≈10⁴ times smaller than the stellar density at the depth ΔR (while the sound speed is by a factor of two smaller).

The geometrical cross-section of the flow can be calculated using equation (1). We get y_s ≈ 0.027, which translates into the flow radius of 2.67 × 10¹⁰ cm and the total cross-section area of 2.24 × 10²¹ cm². We note that ρ v decreases fast with increasing y, especially for a high value of n, which decrease is taken into account in our formulae for either 〈ρv〉 or Σ_eff, equations (8), (11), respectively. For the example considered here, ρv drops to 10⁻³ of its value at the point L₁ at y ≈ 0.0215.

Now, let us compare the values of different parameters of the flow obtained with the help of different formulae used in literature.

We shall start with the geometrical cross-section of the flow. As was shown above, the analysis of PS72 leads to the radius of the flow equal to 2.67 × 10¹⁰ cm. Using estimate of LS75 given by equation (18), we get r_f ≈ 1.89 × 10¹⁰ cm, i.e. a value smaller by a factor of only ≈1.4. Taking into account the approximate character of both estimates, the agreement is excellent. We have to remember, however, that using equation (18) requires the knowledge of the sound speed at the L₁ point. In our case, this knowledge was based on the analysis following the approach of PS72 and on the simultaneous integration of the equations of stellar structure for the outer layers of the donor.

Now, let us compare the values of the flow effective (as opposed to the geometrical one) cross-section area obtained with the different formulae listed in Section 2. The value following from the analysis of PS72 is given by equation (11). For the considered example, Σ_eff ≈ 2.3 × 10²⁰ cm². Then, the approximation neglecting the averaging of the flow over the cross-section proposed by Zdziarski et al. (2007), of equation (13), yields Σ₀ ≈ 1.5 × 10²² cm², i.e. almost two orders of magnitude too much. The estimate by LS75, as given by our equation (20), leads to the value Σ₀ ≈ 3.56 × 10²⁰ cm². This value is only by a factor ≈1.5 greater than the value obtained by us with equation (11), which again means that the agreement between the two approaches is excellent. Still, it requires the knowledge of the sound speed at L₁.

We next compare the values of the density of the gas at L₁ obtained with the different formulae listed in Section 2. The value following from the analysis of PS72 is given by equation (17), which in our case leads to |$\rho _{\rm \mathit{ L}_1}\approx 3.4\times 10^{-10}$| g cm⁻³. The estimate by LS75, as given by our equation (19), leads to |$\rho _{\rm \mathit{ L}_1}\approx 2.20\times 10^{-10}$| g cm⁻³. This is by a factor of only ≈1.5 smaller than the value obtained by us with equation (17). This is the same factor 1.5 as found by us while comparing the different estimates of the effective cross-section, which identity is a consequence of the equivalence of equations (19) and (20).

The value of the density of the gas at L₁ is certainly one of the important parameters of the flow. The fact that the values of this parameter obtained with two different approaches agree so well is certainly encouraging. It also supports the claim that this estimate is not far from the true value.

Discussing the approximate formulae present in the literature, we should devote attention to popular formulae equivalent to our equation (10) but used in improper way, As an example we may recall equation (9) of Zdziarski et al. (2007). Those authors use equation (10) but they replace the effective cross-section area with the geometrical one and the density of the gas at L₁ with the density on the Roche lobe equipotential away from L₁. Since, as we have seen, the effective cross-section area might be overestimated this way by two orders of magnitude and the density of the gas even by four orders of magnitude, such use of equation (10) might lead to an error reaching even six orders of magnitude. It is therefore crucial, while using formulae of this type, to use the proper value of the gas density.

Another example may be found in Zdziarski et al. (2019). While those authors properly estimated |$\dot{M}$| from their evolutionary model, they still used the above simplified treatment to estimate the mass stored within the L₁ flow, and thus the emptying time of that region, which was then used to estimate the variability time-scale related to changing conditions at L₁. This leads to a major overestimate of the expected variability time-scale. Using the proper description of the flow, we can calculate the mass column density, σ, perpendicular to the flow (i.e. the stored in a 1-cm thick slice around L₁). Integrating equation (17), we obtain

$$\begin{eqnarray} \sigma =\pi A^2 \rho _{\rm \mathit{ L}_1}\frac{y_{\rm s}^2}{n+1}. \end{eqnarray}$$

(22)

For the considered example, y_s ≈ 0.027, and σ ≈ 8.4 × 10¹⁰ g cm⁻¹. On the other hand, Zdziarski et al. (2019) used σ = ρ₀Σ₀, which is 4.8 × 10¹⁶ g cm⁻¹, a factor of ≈5 × 10⁵ too much. This then leads to a corresponding overestimate of the emptying time, Δt. Those authors also used the |$\dot{M}$| following from their evolutionary model and approximated the length of that region as ΔR, yielding Δt ≈ 24 yr. If we make the same assumptions as them but use the correct σ, we obtain Δt ≈ 1300 s. Even if we change the assumption about the length of the L₁ element made in Zdziarski et al. (2019) to a more realistic value, we can increase the above Δt by at most two orders of magnitude. Thus, the conclusion of them that Δt can account for the observed long-term changes of the accretion rate average over outburst is incorrect, and another mechanism is needed.

Finally, we should comment on the discrepancy between the density of the gas at the Roche equipotential inside the star far from L₁ obtained in our calculations and the density obtained with the prescription of LS75, equation (21). Using it, we find the ratio of ≈52. As we saw in our analysis above, that ratio was about four orders of magnitude (3.1 × 10⁻⁶ versus 3.4 × 10⁻¹⁰ g cm⁻³). This is a serious discrepancy, by more than two orders of magnitude. However, we believe that our estimate is closer to the real situation than that of LS75. The reason is the following one. We started with the determination of the radius excess, ΔR. We used for that purpose equation (14). Fortunately, the sensitivity of ΔR to |$\dot{M}$| is very weak. As demonstrated by Zdziarski et al. (2019) (see their fig. 3), increasing |$\dot{M}$| by four orders of magnitude leads to increase of ΔR by a factor of only two. Since we have a relatively reliable estimate of |$\dot{M}$| based on evolutionary considerations, we may trust that ΔR is determined with rather good precision. Knowing the depth of the Roche equipotential below the stellar surface, we may integrate the stellar structure equations (including the heat transfer equation) from the stellar surface inward. At the depth ΔR, we determine, among others, the density of the gas and the gas sound speed. This procedure is rather straightforward and should not raise serious doubts. On the other hand, the LS75 estimate is based on rather qualitative analysis of a very complicated flow in the outer stellar layers.

Finally, we should stress that all the considered descriptions are still quite approximate. Hydrodynamic simulations are needed to get more accurate descriptions.

4 CONCLUSIONS

We have overviewed and extended the treatment of the flow through L₁ of PS72, and compared it with other, simplified, methods. We have stressed that the flow density at L₁ can be several orders of magnitude lower than that at the same equipotential but away from L₁.

Furthermore, one of the approximate estimates of the flow effective cross-section used in literature overestimates the actual one by two orders of magnitude. The other one, given by LS75, leads to result that remains in excellent agreement with our estimate. The same is true about the estimate of the gas density at L₁ given by LS75 (but to calculate it correctly we have to know the structure of the outer layers of the donor).

However, we have found that the estimate of the ratio of the density of the gas at the Roche equipotential inside the star far from L₁ to that at L₁ according to the prescription of LS75 is not accurate.

The overestimates of the density and the cross-section in some formulae then can lead to a gross overestimate of the outflow rate (calculated as a function of the parameters of the flow), as well as of the amount of mass present in the L₁ flow. We should, however, remember that outflow rate is determined by the evolutionary engine of the donor.

Generally, we conclude that such simplified formulae can give reasonably accurate estimates if they are supported by simultaneous integrations of the equations of stellar structure for the outer layers of the donor.

ACKNOWLEDGEMENTS

We thank the referees for valuable suggestions. This research has been supported in part by the Polish National Science Centre grants 2015/18/A/ST9/00746 and 2019/35/B/ST9/03944.

DATA AVAILABILITY

There are no new data associated with this article.

REFERENCES

Coriat

M.

,

Fender

R. P.

,

Dubus

G.

,

2012

,

MNRAS

,

424

,

1991

10.1111/j.1365-2966.2012.21339.x

Crossref

Search ADS

Frank

J.

,

King

A.

,

Raine

D.

,

2002

,

Accretion Power in Astrophysics

.

Cambridge Univ. Press

,

Cambridge

, p.

398

Jȩdrzejec

E.

,

1969

,

M.S. thesis

,

Warsaw Univ

,

unpublished

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

Lai

D.

,

Helling

C.

,

van den Heuvel

E. P. J.

,

2010

,

ApJ

,

721

,

923

10.1088/0004-637X/721/2/923

Crossref

Search ADS

Lubow

S. H.

,

Shu

F. H.

,

1975

,

ApJ

,

198

,

383

10.1086/153614

Crossref

Search ADS

Paczyński

B.

,

1967

,

Acta Astron.

,

17

,

287

Paczyński

B.

,

Sienkiewicz

R.

,

1972

,

Acta Astron.

,

22

,

73

Savonije

G. J.

,

1978

,

A&A

,

62

,

317

Savonije

G. J.

,

1979

,

A&A

,

71

,

352

Savonije

G. J.

,

1983

, in

Lewin

W. H. G.

,

van den Heuvel

E. P. J.

, eds,

Accretion-driven Stellar X-ray Sources

.

Cambridge Univ. Press

,

Cambridge

, p.

343

Google Scholar

Google Preview

OpenURL Placeholder Text

WorldCat

Webbink

R. F.

,

Rappaport

S. A.

,

Savonije

G. J.

,

1983

,

ApJ

,

270

,

678

10.1086/161159

Crossref

Search ADS

Zdziarski

A. A.

,

Wen

L.

,

Gierliński

M.

,

2007

,

MNRAS

,

377

,

1006

10.1111/j.1365-2966.2007.11686.x

Crossref

Search ADS

Zdziarski

A. A.

,

Ziółkowski

J.

,

Mikołajewska

J.

,

2019

,

MNRAS

,

488

,

1026

10.1093/mnras/stz1787

Crossref

Search ADS

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Article Contents

A comment on the properties of the matter flow through the first Lagrangian point

ABSTRACT

1 INTRODUCTION

2 ESTIMATES OF THE FLOW PARAMETERS

2.1 The approach of PS72

2.2 The approach of LS75

3 COMPARISON WITH A MODEL OF GX 339–4

4 CONCLUSIONS

ACKNOWLEDGEMENTS

DATA AVAILABILITY

REFERENCES

Citations

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Article Contents

A comment on the properties of the matter flow through the first Lagrangian point Free

ABSTRACT

1 INTRODUCTION

2 ESTIMATES OF THE FLOW PARAMETERS

2.1 The approach of PS72

2.2 The approach of LS75

3 COMPARISON WITH A MODEL OF GX 339–4

4 CONCLUSIONS

ACKNOWLEDGEMENTS

DATA AVAILABILITY

REFERENCES

Citations

Views

Altmetric

Email alerts

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Citing articles via

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A comment on the properties of the matter flow through the first Lagrangian point