Angular momentum loss (AML) mechanisms and dynamical evolution owing to
magnetic braking and gravitational radiation in relativistic binary stars (RBS)
are studied with use of physical parameters collected from the literature. We
have calculated and compared AML time scales for the RBS with non-degenerate
components and double degenerate (DD) systems.Comment: 6 pages, 3 figure

Kalomeni, B.

Tout, C. A.

Yakut, K.

English

arXiv

7th Microquasar Workshop: Microquasars and Beyond, MQW 2008; Foca, Izmir; Turkey; 1 September 2008 through 5 September 2008Angular momentum loss (AML) mechanisms and dynamical evolution owing to magnetic braking and gravitational radiation in relativistic binary stars (RBS) are studied with use of physical parameters collected from the literature. We have calculated and compared AML time scales for the RBS with non-degenerate components and double degenerate (DD) systems.Turkish Scientific and Technical Research Counci

Yakut, Kadri

Kalomeni, Belinda

DSpace@IZTECH

Angular momentum loss by magnetic braking and gravitational radiation in relativistic binary stars

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08 Angular Momentum Loss by Magnetic Braking andGravitational Radiation in Relativistic Binary Stars
K. Yakut
University of Cambridge, Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
University of Ege, Department of Astronomy & Space Sciences, 35100 ˙Izmir, Turkey
E-mail: yakut@ast.cam.ac.uk
B. Kalomeni
˙Izmir Institute of Technology, Department of Physics, 35430 ˙Izmir, Turkey
E-mail: belindakalomeni@iyte.edu.tr
C. A. Tout
University of Cambridge, Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK
E-mail: cat@ast.cam.ac.uk
Angular momentum loss (AML) mechanisms and dynamical evolution owing to magnetic brak-
ing and gravitational radiation in relativistic binary stars (RBS) are studied with use of physical
parameters collected from the literature. We have calculated and compared AML time scales for
the RBS with non-degenerate components and double degenerate (DD) systems.
VII Microquasar Workshop: Microquasars and Beyond
September 1-5 2008
Foça, ˙Izmir, Turkey
c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/
Angular Momentum Loss in Relativistic Binary Stars
1. Introduction
White dwarfs (WD), neutron stars (NS), and black holes (BH) represent the endpoint of stellar
evolution. Compact objects provide conditions to study physics at high energy, magnetic fields,
temperatures etc., and they are dominated by gravitation. Important properties of these compact
objects can be determined if they are members of binary systems. This make possible to determine
the orbital elements and physical parameters from precise data obtained by various techniques.
Such studies are important to test the predictions of stellar evolution theory. Additionally, binary
systems with a compact companion or double degenerate (DD), owing to their general relativistic
effects, are good laboratories to test general relativity.
The literature has been scanned for the physical parameters of High Mass X-Ray Binary
(HMXRB) and Low Mass X-Ray Binary (LMXRB) systems with a black hole (BH) or neutron
star (NS) companion, as well as other systems listed in Table 1. We use the data obtained to study
the angular momentum loss mechanisms and focus on the dynamical evolution driven by magnetic
braking and gravitational radiation in relativistic binaries.
2. Data selection
In modern astrophysics, theoretical and observational studies of compact objects, with both
ground based and satellite observations, offer potential to research the formation and evolution. In
recent years, as a result of the advanced technology and instrumentation, the number of papers on
relativistic binaries has increased. These studies contain significantly accurate orbital elements and
physical parameters. The mass interval of these binary systems are summarized in Table 1. Isolated
black holes, neutron stars and white dwarfs are included for comparison. Using these parameters
collected from the literature we have studied their possible angular momentum loss mechanisms.
3. Angular Momentum Loss
The orbital period of the binary system can be determined accurately. Any orbital period
variation in tidally locked binary systems is due to evolutionary changes or the presence of a third
body orbiting the binary. In close binaries, in general, we can estimate the angular momentum
variation via observations of period changes. Angular momentum loss leads to a period decrease.
It can be studied as the orbital angular momentum and spin angular momentum. The spin angular
momentum (Hs) for a single star is given in the form,
Hsi = Iω = k2R2i Miωi, (3.1)
where, subscript i stands for the components, i = 1 for the primary and i = 2 for the secondary,
respectively, I is the star’s moment of inertia, ω is the spin angular velocity, k2 is the gyroton
radius, Ri is the radius of the star, and Mi is the mass of the component. The total orbital angular
momentum is
Hot = M1M2
√
Ga(1− e2)
M
= 111q(1+ q)−2(
Mt
M⊙
)−5/3(
P
day )
1/3
√
1− e2 (M⊙R2⊙/day) (3.2)
2
Angular Momentum Loss in Relativistic Binary Stars
Table 1: Mass distribution of relativistic binaries and isolated compact stars based observational data set.
component M1 M2
Type pri+sec n max. min. max. min.
HMXRB BH+S 6 23.1 6.0 70 6.5
HMXRB NS+S 11 2.4 1.0 58 8.8
LMXRB BH+S 13 14 4.0 2.7 0.37
LMXRB NS+S 6 1.8 1.4 2.3 0.4
NLCV WD+S/B 39 1.4 0.3 1.1 0.05
mCV WD+S 10 0.8 0.4 0.5 0.1
PreCV WD+S 21 0.84 0.39 0.93 0.1
DDNSNS NS+NS 8 1.6 1.14 1.4 1.05
DDNSWD NS+WD 9 2.1 1.27 1.3 0.16
DDWDWD WD+WD 10 0.7 0.32 0.7 0.29
AMCVn WD+WD 5 0.98 0.59 0.13 0.011
Black Hole 19 23.1 4.0
Neutron Star 42 2.4 1.0
White Dwarf 90 1.4 0.01
where G is gravitational constant, a is the separation of two stars and q is the mass ratio of the
components specify q = M2M1 .
The angular momentum is extracted from the binary orbit because of the (i) magnetic braking
in a magnetized stellar wind (MSW), (ii) gravitational radiation (GR), (iii) non-conservative mass
transfer and (iv) a third (or multi-) body effect. The total system angular momentum loss is the sum
of all these effects. We assume throughout this study that relativistic binary star evolution is driven
by GR and MSW.
3.1 Magnetic Braking in a Magnetized Stellar Wind
The Sun is known to rotate more rapidly at the equator than at the poles. The speed of this
differential rotational and convection are important for the production of magnetic fields in the
convective zone (see Yakut & Eggleton and references therein). Light curves of close binary stars
show one maximum higher than the other. This phenomenon is called the O’Connell effect (Milone
1968). These asymmetries are usually attributed to the magnetic activity, though the effect is not
satisfactorily explained by existing the models. A large number of late-type stars show magnetic
activity indicators. Hence, activity may play an important role in relativistic binaries whose com-
ponents are late-type stars (M . 1.5 M⊙) .
The Skumanich relation (Skumanich 1972, Smith 1979 ) predicts a strong correlation between
equatorial rotation velocity and age as
ve = 1014× f × t−0.50
cm s−1, where t0 is the age of the star and the value of f was determined empirically by Skumanich
(1972) to be 0.7 and Smith (1979) to be 1.78. The Skumanich law and Eq. (3.1) yield the angular
momentum loss by MSW,
(
dH
dt
)
MSW
=−1.6×10−30M2R42ω3. (3.3)
3
Angular Momentum Loss in Relativistic Binary Stars
The angular momentum loss time scale
τMSW = 14(
M
M⊙
)2/3(
R2
R⊙
)−4(
P
day)
10/3(1+ q)−1(1− e2)1/2×Gyr. (3.4)
If the secondary stars in binaries fill their Roche lobes then the radius may be calculated with
the relation given by Eggleton (1983),
R2 = RL = a
0.49q2/3
0.6q2/3 + ln(1+ q1/3)
. (3.5)
Otherwise we used the mass-radius relations
R = M0.97 for M2 ≤ 0.5M⊙ and R = M1.11 for 0.5M⊙ < M2 ≤ 1.5M⊙ (3.6)
given by Yakut et al. (2008a).
3.2 Gravitational Radiation
Binary star systems, in which both components are white dwarfs, neutron stars, or black holes
emit gravitational waves that can carry off angular momentum. The shape of the binary orbit and
the energy of the system given by
r =
a(e2−1)
1+ ecosν
and E = −GM1M2
2a
. (3.7)
These equations and the semi-major axis variation rates (Landau & Lifshitz, 1951) yield the angular
momentum loss by the gravitational radiation and the angular momentum loss time scale (in Gyr)
da
dt =−
64
5
G3
c5
M1M2
a3
(M1 + M2), (3.8)
(
dH
dt
)
GR
=−3.44×1044(M1M2M−1/3)2P−7/3 f (e) (3.9)
τGR = 376.4q−1(1+ q)2M−5/3P8/3(1− e2)3/2(1+
7
4
e2)−1Gyr. (3.10)
3.3 Timescales
The angular momentum loss time scales for relativistic and non-relativistic binary stars (LTCB
and HTCB) are shown in Figs.1 and 2. The details will be discussed in Yakut et al. (2008b). Fig.
3 is the same as Figs.1 and 2 but for double compact degenerate binary systems.
4. Conclusion
The aim is to collate the physical properties of the relativistic binary stars because they are
valuable in studies of formation and evolution of compact objects. From observational data (see
Table 1) we also summarize mass distribution for the compact objects. According to the data
set, maximum value for mass of black holes, neutron stars and white dwarfs are 23, 2.4 and 1.4
4
Angular Momentum Loss in Relativistic Binary Stars
Figure 1: Plot of logτMSW vs. logP. The open circles, asterisks, and filled circles show binaries with WD,
NS and BH companions, respectively.
Figure 2: Plot of logτGR vs. logP. The open circles, asterisks, filled circles show binaries with WD, NS,
and BH companions and + and x show the LTCB and HTCB systems, respectively.
5
Angular Momentum Loss in Relativistic Binary Stars
Figure 3: Plot of logτGR vs. logP for double degenerate binaries. The open circles, open squares, filled
circles and triangles indicate WD+WD, WD+WD (AMCVn), NS+NS and NS+WD binaries, respectively.
M⊙, respectively and for minimum mass for black holes, neutron stars and white dwarfs are 4, 1,
and 0.01 respectively. We also draw attention to gap between minimum black hole masses and
maximum mass of neutron stars (see for details Yakut et al. 2008b). The data obtained have
been used to study the angular momentum loss and its evolution driven by magnetic braking and
gravitational radiation. Binary stars with BH components have longer orbital periods them those
with WD and NS components. Increase in the orbital period increases the efficiency of AML by
GR. Greater the difference between the mass ratio the shorter is the AML time-scale via MSW.
Acknowledgments
KY and BK acknowledges support by the Turkish Scientific and Technical Research Council
(TÜB˙ITAK).
References
[1] Eggleton, P. P., 1983, ApJ, 268, 368
[2] Landau, L. D.& Lifshitz, E. M., The classical theory of Fields, 1951, Pergamon Press, Oxford.
[3] Milone E. F., 1968, AJ, 73, 708
[4] Skumanich, A. 1972, ApJ, 171, 565
[5] Smith, M. A., 1979, PASP, 91, 737
[6] Yakut, K. & Eggleton, P.P., 2008, ApJ, 629, 1055
[7] Yakut, K., et al. 2008a, to be submitted
[8] Yakut, K., et al. 2008b, to be submitted
6


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