Download Non-conservative mass transfers in Algols

MNRAS 441, 1166–1176 (2014)
doi:10.1093/mnras/stu630
Non-conservative mass transfers in Algols
A. Erdem1,2‹ and O. Öztürk1,2
1 Astrophysics
2 Department
Research Centre and Observatory, Çanakkale Onsekiz Mart University, Terzioğlu Kampüsü, TR-17020 Çanakkale, Turkey
of Physics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart University, Terzioğlu Kampüsü, TR-17020 Çanakkale, Turkey
Accepted 2014 March 30. Received 2014 February 10; in original form 2013 October 26
ABSTRACT
We applied a revised model for non-conservative mass transfer in semi-detached binaries to
18 Algol-type binaries showing orbital period increase or decrease in their parabolic O−C
diagrams. The combined effect of mass transfer and magnetic braking due to stellar wind was
considered when interpreting the orbital period changes of these 18 Algols. Mass transfer was
found to be the dominant mechanism for the increase in orbital period of 10 Algols (AM Aur,
RX Cas, DK Peg, RV Per, WX Sgr, RZ Sct, BS Sct, W Ser, BD Vir, XZ Vul) while magnetic
braking appears to be the responsible mechanism for the decrease in that of 8 Algols (FK Aql,
S Cnc, RU Cnc, TU Cnc, SX Cas, TW Cas, V548 Cyg, RY Gem). The peculiar behaviour of
orbital period changes in three W Ser-type binary systems (W Ser, itself a prototype, RX Cas
and SX Cas) is discussed. The empirical linear relation between orbital period (P) and its rate
of change (dP/dt) was also revised.
Key words: binaries: close – binaries: eclipsing..
1 I N T RO D U C T I O N
In general, classical Algol-type binary stars consist of an early-type
main-sequence star as the primary component and a late-type subgiant or giant star as the secondary component (e.g. Kopal 1955;
Giuricin, Mardirossian & Mezzetti 1983; Peters 2001). They are
generally semi-detached (SD) binary systems. In this type of eclipsing binary, the second component fills its Roche lobe, after which
mass transfer from the secondary component to the primary component occurs. Since the secondary components are of late spectral
type, these components are magnetically active and produce spots,
plages, flares and winds (e.g. Hall 1989). They lose angular momentum due to magnetic stellar winds; therefore, mass transfer
between the components of Algol-type binary systems should be
non-conservative (e.g. Sarna, Muslimov & Yerli 1997; Sarna, Yerli
& Muslimov 1998).
An extreme example of this type of mass transfer mechanism is
seen in W Ser-type binary systems (e.g. Plavec 1980, 1989; Wilson
1989). Since the secondary component is generally a late-type giant
star, a large amount of mass is rapidly transferred from the secondary
component to the early-type primary component. Thus, an optically
thick shell or disc occurs around the primary component so that
the shell or disc hides the hot component and obscures its light and
spectral lines.
In SD binaries, it is known that conservative mass transfer from
the Roche-lobe-filling, less massive, secondary component to the
more massive primary component causes an increase in orbital
E-mail: [email protected]
period. However, the magnetic braking process of stellar winds
can cause a decrease in orbital period. Therefore, the net effect of
non-conservative mass transfer (in terms of stellar wind mechanism
+ mass transfer) should be considered when interpreting orbital
period changes in classic Algols (Qian 2000a,b, 2001a,b,c; Yang &
Wei 2009; Erdem et al. 2010; Soydugan et al. 2011).
On the other hand, the orbital and stellar spin evolution in Algoltype binary stars has been investigated by many authors (more recently İbanoğlu et al. 2006; Dervişoğlu, Tout & İbanoğlu 2010; van
Rensbergen et al. 2010, 2011; Deschamps et al. 2013). Many physical mechanisms (conservative and non-conservative mass transfer,
torques, tidal effects, disc accretion, magnetic braking, etc.) were
taken into account for modelling Algol evolutions. However, it appears that more accurate and detailed observational data are needed
in order to test their models.
A model of non-conservative mass transfer in SD binaries, proposed by Tout & Hall (1991) and Erdem et al. (2005), is considered here. Making appropriate adjustments, the revised model was
applied to 18 Algol-type binaries selected from the catalogue of
Algol-type binary stars presented by Budding et al. (2004), including only those with parabolic O−C diagrams, indicating orbital
period change.
2 PERIOD CHANGE DUE TO MASS
TRANSFER/LOSS
In a binary system, the total angular momentum of the binary is
composed of the sum of the orbital angular momentum and the
rotational, or spin, angular momentum of each component star.
In practice, since the total spin angular momentum is less than
C 2014 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Non-conservative mass transfers in Algols
1167
1–2 per cent of the total orbital angular momentum in most cases, it
is possible to ignore the spin angular momentum of the components
altogether and to write the following equation for a close binary
system with a circular orbit:
angular momentum of the system should be constant(Ṁ = 0 and
L̇ = 0). Then we have
L ≈ Lorb. = M1 d12 ω + M2 d22 ω,
where Ṁ 1 > 0 and Ṁ 2 < 0. Under these assumptions, from equation (7), we obtain
(1)
where M1 and M2 are the masses of the primary and secondary
component stars, d1 and d2 are the distances of the respective components from the centre of mass of the binary system, and ω is the
angular velocity of the component stars about the centre of mass of
the binary system.
The binary system is balanced due to torque forces so that
M1 d1 = M2 d2 .
(2)
If the separation between the two components is d (d = d1 + d2 ),
then
M2
M1
d and d2 =
d.
(3)
d1 =
M
M
Equations (1) and (3) can be combined to write
M1 M2 2
d ω.
(4)
M
On the other hand, Kepler’s third law for a binary system with a
circular orbit is given by
L=
G
d3
=
(M1 + M2 ),
P2
4π2
(5)
where P (ω = 2Pπ ) is the orbital period of the binary system, G is
the universal gravitational constant and M is the total mass of binary
system.
These last two equations can be combined to give
2 1/3
G
M1 M2 1/3
P .
(6)
L=
2π
M 1/3
The time derivative of equation (6) gives
Ṁ
Ṁ 1
Ṁ 2
L̇
Ṗ
=
−3
−3
+3 .
P
M
M1
M2
L
Ṁ 2 = −Ṁ 1 ,
(8)
3(M2 − M1 )
Ṗ
=
Ṁ 2 .
P
M1 M2
(9)
This equation represents the orbital period change due to mass
transfer from the secondary to the primary component. Since the
secondary component is less massive than the primary component in
classic Algol-type binaries, the conservative mass transfer in these
systems always produces an orbital period increase.
2.2 Non-conservative mass transfer
If we assume that the evolved secondary component is losing mass
at a rate of −Ṁ 2 and that some of this matter is transferred to the
primary component at a rate of Ṁ 1 , while the rest is lost from the
system at a rate of Ṁ, then we have
Ṁ 2 = Ṁ − Ṁ 1 ,
(10)
where Ṁ 1 > 0, Ṁ 2 < 0 and Ṁ < 0. The binary system loses angular momentum due to escaping matter and other mechanisms (such
as strong stellar wind). Under these assumptions, from Tout & Hall
(1991):
L̇ = Ṁd22 ω + KL,
(11)
where KL represents additional angular momentum loss due to any
other mechanism. Equations (4) and (11) can therefore be combined
to write
M1 Ṁ
L̇
=
+ K.
L
M2 M
(12)
Substituting this equation for angular momentum loss into equation (7) gives
(7)
Equation (7) represents the general effect of changes in mass
and total angular momentum on the orbital period of a binary system. It is well known that in SD binary stars the Roche-lobe-filling
component transfers mass to the other (accreting) component. The
mass transfer dominates the evolution of the binary system. In many
cases, the inflowing matter forms an accretion disc around the accreting star. On the other hand, a binary system may lose angular
momentum due to strong stellar wind. The period variation can be
derived from observations, while equation (7) has four unknown
quantities. We shall now discuss the general effect of mass transfer
and angular momentum loss on the orbital period of the SD binary
system.
2.1 Conservative mass transfer
In classic SD binaries (such as Algol-type binaries), the less massive secondary component fills its Roche lobe while the volume of
the more massive primary component is less than its Roche lobe
volume. Therefore, the secondary component transfers mass to the
primary component through the inner Lagrangian point. We may
suppose that the secondary component is losing mass at a rate of
−Ṁ 2 and the primary component is gaining mass at a rate of Ṁ 1 .
In the case of conservative mass transfer, the total mass and total
Ṗ
2
3(M1 − M2 )
= − Ṁ +
Ṁ 1 + 3K.
P
M
M1 M2
(13)
There are two main causes of additional angular momentum loss
in a binary: magnetic braking and gravitational-wave radiation.
For magnetic braking (or Alfvén-driven mass transfer), from Tout
& Hall (1991):
2
ṀRA2 ω
3
or, with equation (4),
2 RA 2 M
Ṁ.
K=
3
d
M1 M2
KL =
(14)
(15)
Here RA is the Alfvén radius, inside which matter lost from the
system is corotating with the magnetically active component star.
In practice, a reasonable upper limit for the Alfvén radius might be
10 stellar radii (see Tout & Hall 1991).
Substituting this expression for additional angular momentum
loss by magnetic braking into equation (13) provides
RA 2 M
2
3(M1 − M2 )
Ṗ
Ṁ 1 .
−
(16)
= 2
Ṁ +
P
d
M1 M2
M
M1 M2
This final equation relates the time derivative of the orbital period
to the rate of mass transfer and loss in SD systems (such as Algols)
MNRAS 441, 1166–1176 (2014)
1168
A. Erdem and O. Öztürk
under the hypothesis of Alfvén-driven mass transfer. The first term
on the right-and side of equation (16), which is always negative,
represents the mass-loss effect on the period change. The second
term, which is always positive for classical Algols, represents the
period increase due to mass transfer.
On the other hand, the parameter β, giving the fraction of mass
lost by the secondary component that is accreted by the primary
component, could be considered as follows:
Ṁ 1 = −β Ṁ 2 ,
(17)
β = 1. Substituting this equation for mass-loss into equation (10)
gives
Ṁ ≤ (1 − β) Ṁ 2 =
(β − 1)
Ṁ 1 .
β
(18)
Ṗ
= c1 Ṁ + c2 Ṁ 1 ,
P
where
RA 2
M
2
− ,
c1 = 2
d
M1 + M2
M
(19)
3 (M1 − M2 )
.
M1 M2
Comparing equations (18) and (19), the following equation for the
critical value of the mass-loss parameter β and following results
can be derived.
c2 =
(i) If non-conservative mass transfer is the dominant mechanism
and then the orbital period of the binary system increases,
c1
.
β>
c1 +c2
(ii) If mass-loss from the system is the dominant mechanism and
then the orbital period of the binary system decreases,
c1
β<
.
c1 +c2
3 TA R G E T S TA R S A N D O B S E RVAT I O N A L
DATA
The catalogue of Algol-type binary stars prepared by Budding et al.
(2004) provides information on 411 classical Algol systems. The
O−C diagrams of 256 of these Algols are in the atlas of O−C
diagrams of eclipsing binary stars presented by Kreiner, Kim & Nha
(2001). In this study, 18 of these Algols, showing only parabolic
O−C diagrams, were selected in order to investigate their orbital
period changes and apply the revised model of non-conservative
mass transfer.
The SD parameter in Table 1 is a crude probability indicator for
binaries having an SD nature used by Budding (1984) and Budding
et al. (2004). Here, this parameter takes five values from 0.1 to
0.9 in steps of 0.2: 0.9 for well-known SD cases; 0.7 for binaries of
apparently similar properties to 0.9 cases (though generally less well
known); 0.5 for binaries where an SD or near-contact designation is
about equally likely; and 0.3, indicating rather unlikely SD cases.
The majority of the observational eclipse timings (minima times)
for these 18 Algol-type binaries were obtained from the list compiled by Kreiner et al. (2001). Adding some new minima times
published in the literature, the updated data, summarized in Table 2,
were used and can be requested in electronic or printed format from
the authors.
Almost all observational data were used, although the earlier data
(mostly visual, photographic and photovisual) have relatively less
accuracy. Disregarding these, however, would result in a loss of
information when the system was only able to be observed visually,
photographically and/or photovisually, especially in early epochs.
Since most of the older times of minima have no published standard
errors, the individual times of minima were weighted according to
the type of observations rather than their standard errors. Therefore,
since the visual and photovisual (weak image on photographic plate)
minima times and photographic (photographic series of exposure)
minima times were published to two or three decimal places, their
weights were chosen to be 1 and 5, respectively. On the other hand,
Table 1. Physical parameters of the 18 Algol-type eclipsing binaries studied.
System
Spectral type
Separation
(R )
M1
(M )
M2
(M )
R1
(R )
R2
(R )
SD
Ref.
FK Aql
AM Aur
S Cnc
RU Cnc
TU Cnc
RX Cas
SX Cas
TW Cas
V548 Cyg
RY Gem
DK Peg
RV Per
WX Sgr
RZ Sct
BS Sct
W Ser
BD Vir
XZ Vul
B9+G5III
A8+[G6IV]
B9.5V+G8-9III–IV
F9V+[K0IV]
A0+[G8IV]
A2-5+K0II–III
B7+K3III
B9V+[F6IV]
A0V+[F9IV]
A2+K0-3IV–V
A0V+A2V
A2+[G7IV]
A1+[G4IV]
B2II+A0II–III
B7e-A0III
cF5ep
A5+[K0IV]
F5–8
16.94
30.20
26.36
23.42
18.26
45.37
90.66
8.32
10.66
25.00
9.66
10.05
9.96
62.43
17.88
47.29
11.31
12.81
6.29
1.65
2.51
1.15
2.40
0.62
4.00
2.65
3.90
2.04
2.40
3.04
2.25
11.70
3.26
5.50
2.22
1.85
2.20
0.35
0.23
0.52
0.25
0.58
3.50
1.15
1.10
0.39
2.15
0.46
0.68
2.49
2.01
1.60
0.78
1.11
3.29
2.35
2.15
1.80
2.20
11.38
3.50
2.43
3.70
2.38
2.20
2.91
2.10
15.80
2.73
8.48
2.72
4.12
5.75
5.55
5.25
5.15
3.60
7.26
32.00
2.40
2.70
6.19
2.00
4.52
2.70
15.90
3.54
17.80
4.38
4.85
0.7
0.7
0.9
0.7
0.7
0.5
0.7
0.3
0.5
0.9
0.7
0.5
0.7
0.9
0.7
0.5
0.5
0.3
(1)
(2)
(3)
(2)
(2)
(1)
(4)
(5)
(6)
(7)
(2)
(1)
(2)
(8)
(1)
(9), (10)
(1)
(1)
(1) Brancewicz & Dworak (1980), (2) Svechnikov & Kuznetsova (1990), (3) van Hamme & Wilson (1993),
(4) Gurzadyan (1997), (5) Narita, Schroeder & Smith (2001), (6) Mardirossian et al. (1980), (7) Sarma &
Vivekananda (1997), (8) Olson & Etzel (1994), (9) Piirola et al. (2005), (10) Sanad & Bobrowsky (2013).
MNRAS 441, 1166–1176 (2014)
Non-conservative mass transfers in Algols
1169
Table 2. Summary of observational eclipse timings for the 18 Algol-type eclipsing binaries studied.
System
Data interval
FK Aql
AM Aur
S Cnc
RU Cnc
TU Cnc
RX Cas
SX Cas
TW Cas
V548 Cyg
RY Gem
DK Peg
RV Per
WX Sgr
RZ Sct
BS Sct
W Ser
BD Vir
XZ Vul
1929–2010
1907–2004
1848–2008
1911–2006
1928–2008
1903–2009
1906–1999
1901–2011
1942–2010
1908–2006
1926–2008
1906–2009
1903–2004
1911–2005
1909–1996
1900–2001
1927–2011
1930–2010
Visual
Min I Min II
79
33
133
35
32
47
39
151
119
21
61
135
36
31
28
21
23
41
–
–
–
–
–
19
2
–
–
–
–
–
–
–
–
–
–
–
Photovisual
Min I
Min II
–
49
71
8
12
14
5
3
22
3
1
–
7
9
7
23
–
6
Photographic
Min I Min II
–
–
–
–
–
9
–
–
–
–
–
–
–
–
–
–
–
–
since the CCD and photoelectric minima times were given to four
or five decimal places, their weights were chosen to be 10.
2
8
6
–
12
4
17
35
5
2
2
11
1
7
3
2
1
8
CCD and Photoelectric
Min I
Min II
–
–
–
–
–
1
–
–
–
–
–
–
–
–
–
–
–
–
4
3
13
3
4
28
26
98
51
21
25
22
1
4
1
31
6
3
–
–
–
–
–
11
2
1
2
–
–
–
–
–
–
–
–
–
Total
85
93
223
46
60
133
91
288
199
47
89
168
45
51
39
77
30
58
by Kreiner et al. (2001). In order to check whether the symmetric
parabola at the vertex with E (epoch number) was = 0, the O−C
values of the other eight Algols (AM Aur, S Cnc, TU Cnc, SX Cas,
TW Cas, RU Cnc, DK Peg, RV Per) were computed using the linear
ephemeris given in Tables 3 and 4. The O−C diagrams, constructed
from all available times of eclipse minima (Figs 1–18), show that
the orbital period variations of these binary systems have parabolic
form (upward parabolas or downward parabolas). The weighted
least-squares method was applied to each O−C data set of the 18
4 R E S U LT S A N D D I S C U S S I O N
The O−C method was used to study the orbital period variations
of the selected 18 Algols. The O−C values of 10 Algols (FK Aql,
RX Cas, V548 Cyg, RY Gem, WX Sgr, RZ Sct, BS Sct, W Ser,
BD Vir, XZ Vul) were calculated using the linear ephemeris given
Table 3. Results of O−C analysis of 10 Algol systems showing orbital period increase.
System
T0
240 0000 + HJD
P
(d)
Q × 10−10
(d)
Ṗ
(s yr−1 )
Ṁ 2,con (M yr−1 )
βcri
Ṁ 1
(M yr−1 )
Ṁ (M yr−1 )
AM Aur
RX Cas
DK Peg
RV Per
WX Sgr
RZ Sct
BS Sct
W Ser
BD Vir
XZ Vul
373 63.15(6)
468 27.6(2)
390 29.32(1)
351 71.277(2)
455 18.54(9)
370 33.72(3)
401 48.620(9)
193 21.7(4)
425 38.41(3)
459 32.58(6)
13.6179(1)
32.3330(6)
1.631 812(2)
1.975 3102(3)
2.129 22(1)
15.190 52(5)
3.821 021(3)
14.160(9)
2.548 536(8)
3.089 68(2)
4609(701)
117 368(4078)
10.5(6)
3.1(2)
59(3)
2448(610)
29(5)
42 100(956)
70(5)
147(7)
2.1(3)
22.9(8)
0.041(3)
0.0100(8)
0.174(9)
1.0(3)
0.048(9)
18.8(4)
0.17(1)
0.30(2)
2.69 × 10−7
2.46 × 10−5
1.98 × 10−6
1.05 × 10−8
3.09 × 10−7
8.17 × 10−7
2.54 × 10−7
1.15 × 10−5
3.16 × 10−7
1.04 × 10−6
0.768
0.979
0.980
0.948
0.899
0.867
0.912
0.944
0.954
0.974
10−5 −10−7
10−4 −10−5
10−5 −10−6
10−7 −10−8
10−5 −10−7
10−4 −10−6
10−5 −10−7
10−3 −10−5
10−5 −10−7
10−5 −10−6
10−5 −10−10
10−5 −10−8
10−6 −10−9
10−8 −10−11
10−6 −10−10
10−5 −10−9
10−6 −10−10
10−5 −10−8
10−6 −10−10
10−6 −10−9
Table 4. Results of O−C analysis for eight Algol systems showing orbital period decrease.
Ṁ System
T0
240 0000 + HJD
P
(d)
Q × 10−10
(d)
Ṗ
( s yr−1 )
β cri
Ṁ 1
(M yr−1 )
(M yr−1 )
FK Aql
S Cnc
RU Cnc
TU Cnc
SX Cas
TW Cas
V 548 Cyg
RY Gem
377 86.070(4)
238 58.453(6)
362 31.62(3)
359 66.52(1)
339 63.25(5)
354 42.3834(9)
444 56.497(3)
397 32.65(1)
2.650 880(1)
9.484 531(4)
10.172 93(2)
5.561 455(5)
36.5667(2)
1.428 3238(1)
1.805 2392(7)
9.300 58(1)
−17(1)
−65(12)
−176(55)
−87(15)
−15 159(4201)
−1.0(1)
−7.2(9)
−862(64)
−0.040(5)
−0.043(8)
−0.11(3)
−0.10(2)
−2.6(7)
−0.0045(5)
−0.025(3)
−0.58(4)
0.940
0.757
0.891
0.757
0.992
0.932
0.881
0.855
10−5 −10−11
10−6 −10−12
10−6 −10−12
10−6 −10−12
10−5 −10−11
10−6 −10−12
10−6 −10−11
10−5 −10−11
10−7 −10−8
10−7 −10−9
10−7 −10−9
10−6 −10−9
10−7 −10−8
10−7 −10−9
10−6 −10−8
10−6 −10−8
MNRAS 441, 1166–1176 (2014)
1170
A. Erdem and O. Öztürk
Figure 1. O−C diagram of AM Aur.
Figure 3. O−C diagram of DK Peg.
Figure 2. O−C diagram of RX Cas.
Algol systems to find their quadratic ephemerides. The resulting
parameters and their standard deviations are given in Tables 3 and
4. The observational O−C data and theoretical best-fitting curves,
and also the residuals, are plotted against the epoch number and observation years in Figs 1–18. Some points shown in the elliptic areas
on the O−C diagrams were ignored due to their large scatter. For
S Cnc, TW Cas and V548 Cyg, due to having sufficient CCD and
photoelectric minima times after the 1960s, 1965 and 1985, respectively, those visual, photovisual and photographic points with large
MNRAS 441, 1166–1176 (2014)
Figure 4. O−C diagram of RV Per.
errors were discarded before analysis. The combined effect of mass
transfer and mass-loss via magnetic braking (i.e. non-conservative
mass transfer) was considered when interpreting these long-term
parabolic period variations.
Non-conservative mass transfers in Algols
Figure 5. O−C diagram of WX Sgr.
Figure 7. O−C diagram of BS Sct.
Figure 6. O−C diagram of RZ Sct.
Figure 8. O−C diagram of W Ser.
4.1 Algols with increasing orbital periods
The O−C diagrams of 10 of these 18 Algols show upward parabolic
forms. These 10 Algols are AM Aur, RX Cas, DK Peg, RV Per,
WX Sgr, RZ Sct, BS Sct, W Ser, BD Vir and XZ Vul. Upward
parabolic O−C variations in eclipsing binaries indicate an increase
in their orbital periods. According to Table 3, two interesting Algollike binary systems, which show a very rapid increase in their periods, are RX Cas and W Ser: 22.9 ± 0.8 and 18.8 ± 0.4 s yr−1 ,
respectively. The Algol-type system which exhibits the slowest increase in its orbital period is RV Per: 0.0100 ± 0.0008 s yr−1 .
In Table 3, the rate of conservative mass transfer (Ṁ 2,con ) from the
Roche-lobe-filling secondary component to the primary component
was calculated from equation (9). According to our computations,
1171
RX Cas and W Ser have the highest conservative mass transfer
rate, at 10−5 M yr−1 , while the other mass transfer rates given in
Table 3 are generally of the order of 10−7 M yr−1 . On the other
hand, critical values of the mass-loss parameter (βcri ) were derived
from equation (20) under the assumption that the RA Alfvén radius
is 10R2 corotating distance from the evolved cooler secondary star
(see Tout & Hall 1991). As mentioned in Section 2.2, if the massloss parameter is larger than its critical value, the non-conservative
mass transfer is the dominant mechanism and the orbital period of
the binary system then increases. Therefore, the range of possible
values of the mass gain rate (Ṁ 1 ) of the primary component and the
mass-loss rate (Ṁ) of the system in Table 3 were estimated from
equations (18) and (19) for βcri < β < 1. The change of mass-loss
MNRAS 441, 1166–1176 (2014)
1172
A. Erdem and O. Öztürk
Figure 9. O−C diagram of BD Vir.
Figure 11. O−C diagram of FK Aql.
Figure 10. O−C diagram of XZ Vul.
parameter β versus the mass-loss rate, Ṁ 2 , of the secondary component of W Ser is shown in Fig. 19 as an example.
W Ser, itself a prototype of one type of variable stars, is a highly
interacting eclipsing binary system with an orbital period of 14.16 d.
The mass-gaining primary component could be a B-type star embedded in a geometrically and optically thick disc (Plavec 1989),
while the mass-losing secondary component is a giant F5 II star
(e.g. Young & Snyder 1982). There is considerable observational
evidence of rapid mass transfer and extensive circumstellar matter
in W Ser. For example, Weiland et al. (1995) made spectroscopic
observations of W Ser, obtained at two of its orbital phases, using the
Goddard High Resolution Spectrograph aboard the Hubble Space
Telescope. They proposed a model of an accretion disc surrounding
the primary component, which is geometrically and optically thick,
using strong absorption features superimposed on the Si IV emission
MNRAS 441, 1166–1176 (2014)
Figure 12. O−C diagram of S Cnc.
lines. Their model also includes a hotspot on the accretion disc,
which could be a sign of rapid mass transfer from the giant stellar
companion to the mass-gaining hot component. On the other hand,
RX Cas is a W Ser-type eclipsing binary system with an orbital
period of 32.33 d. According to Kalv (1979), the mass-gaining primary component is a B-type star embedded in an optically opaque
thick accretion disc and the mass-losing secondary component is a
G3 giant star. There is much photometric and spectroscopic observational evidence for rapid mass transfer and a dense circumstellar
Non-conservative mass transfers in Algols
Figure 13. O−C diagram of RU Cnc.
1173
Figure 15. O−C diagram of SX Cas.
Figure 14. O−C diagram of TU Cnc.
Figure 16. O−C diagram of TW Cas.
envelope in the system (e.g. Kalv 1979; Plavec 1980; Todorova
& Khruzina 1989; Todorova 1993). Therefore, the observed high
values of the orbital period increase in these two systems could be
caused by rapid and large mass transfer from the giant component
to the primary component.
RV Per, among the 18 Algols studied in this paper, shows the
slowest increase in its orbital period. The spectral type of the system was classified as A2+G7IV by Svechnikov & Kuznetsova
(1990). However, no emission lines, which indicate existence of an
MNRAS 441, 1166–1176 (2014)
1174
A. Erdem and O. Öztürk
Figure 19. Change of mass-loss parameter (β) versus mass-loss rate (Ṁ 2 )
of the secondary component of W Ser.
4.2 Algols with decreasing orbital periods
Figure 17. O−C diagram of V548 Cyg.
Figure 18. O−C diagram of RY Gem.
accretion disc in the system, were found in the study of Kaitchuck,
Honeycutt & Schlegel (1985). Zasche (2008) accounted for the
orbital period change of the system as being a light-time effect
due to a third body. However, as mentioned by Zasche (2008),
the orbital period of a third body is not covered by observations in the O−C diagram of RV Per, the errors of the light-time
effect parameters are high and there are no observations near the
periastron passage of the third body, leaving a third-body hypothesis in doubt. The last century of data from RV Per were
found to be insufficient to decide on the character of the secular O−C variation: parabolic or sinusoidal. Future observations
over the next two decades may help determine the character of the
variation.
MNRAS 441, 1166–1176 (2014)
The O−C diagrams of eight Algols show downward parabolic
forms. These eight Algols are FK Aql, S Cnc, RU Cnc, TU Cnc,
SX Cas, TW Cas, V548 Cyg and RY Gem. Downward parabolic
O−C variations in eclipsing binaries indicate a decrease in their
orbital periods. According to Table 4, SX Cas has the most rapid
decrease, at a rate of −2.6 ± 0.7 s yr−1 in its orbital period, while TW
Cas shows the slowest decrease, at a rate of −0.0045 ± 0.0005 s yr−1
in its orbital period.
As mentioned in Section 2.2, if the orbital period of a classical
Algol-type binary system decreases, the mass transfer from the
Roche-lobe-filling secondary component to the primary component
should be non-conservative. Therefore, only the non-conservative
mass transfer is considered in this subsection. Critical values of the
mass-loss parameter (β cri ) were derived from equation (20) under
the assumption that the RA Alfvén radius is 10R2 corotating distance
from the evolved cooler secondary star (see Tout & Hall 1991).
As per 2.2, if the mass-loss parameter is smaller than its critical
value, mass-loss from the system via stellar wind is the dominant
mechanism and the orbital period of the binary system subsequently
decreases. Therefore, the range of possible values of the mass gain
rate (Ṁ 1 ) of the primary component and the mass-loss rate (Ṁ) of
the system in Table 3 were estimated from equations (18) and (19)
for 0 < β < β cri . The change in mass-loss parameter, β, versus the
mass-loss rate, Ṁ 2 , of the secondary component of SX Cas is given
in Fig. 20 as an example.
SX Cas is also a W Ser-type eclipsing binary system with a long
orbital period of 36.57 d. This system has an SD configuration: its
mass-gaining primary component is a B7 star surrounded by a geometrically and optically thick disc and its mass-losing secondary
component is a giant K3III star (e.g. Andersen et al. 1988; Plavec,
Weiland & Koch 1982). There is also plentiful photometric, polarimetric and spectroscopic observational evidence for a rapid mass
transfer and a dense/scattering circumstellar envelope in the system
(e.g. Pfeiffer & Koch 1973, 1977; Plavec et al. 1982; Pavlovski
& Kriz 1985; Andersen et al. 1988; Piirola et al. 2006). Among
these authors, Plavec et al. (1982) found very strong ultraviolet
emission lines (C IV 1550, Si IV 1400, N V 1240 Å, etc.) in the
IUE and optical spectra of the system, covering the wavelength
Non-conservative mass transfers in Algols
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5 S U M M A RY A N D C O N C L U S I O N
A model for non-conservative mass transfer in SD binaries was
revised under the hypothesis of Alfvén wave-driven mass transfer.
18 Algol-type binaries with parabolic O−C diagrams were selected
as target stars to apply this model from the catalogue of Algol-type
binary stars compiled by Budding et al. (2004). The results can be
summarized as following.
Figure 20. Change of mass-loss parameter (β) versus mass-loss rate (Ṁ 2 )
of the secondary component of SX Cas.
interval 110ñ 680 nm. Gurzadyan (1997) proposed that a white
dwarf with an accretion disc (as a component of a local binary
system with a B7 primary star) could be responsible for these strong
UV emission lines. On the other hand, using the multiwave band
(UBVRI) linear polarization data of the system, Piirola et al. (2006)
developed a model in which the mass-gaining primary component
of SX Cas is less obscured by circumstellar matter than that of W
Ser. Furthermore, they did not find any clear evidence for a spherical
shell around the primary or mass outflow in the polar directions.
According to these results, the highest observed value of the orbital
period decrease of SX Cas could be caused by the high mass-loss
rate of 10−7 M yr−1 from the system under the assumption of a
transferred mass 10−8 M yr−1 .
S Cnc is a long-period (≈9.4845 d) detached Algol-like binary with a secondary component, which is close to filling its
Roche lobe (see Olson & Etzel 1993). Spectroscopic and photometric observations of the system were performed by many authors
(e.g. Popper & Tomkin 1984; Etzel & Olson 1985; Olson & Etzel
1993; Hayasaka 2000; Glazunova et al. 2008; İbanoğlu et al. 2012).
The spectral type of the components of the system is given as
B9.5V + G8-K0 III-IV (Etzel & Olson 1985; Olson & Etzel 1993).
According to Glazunova et al. (2008), the rotational velocity of
the primary component differs from the synchronous rotation by
more than a factor of 2; the possible reason for this asynchronous
rotation is most likely the high mass transfer rate. They suggest
that since S Cnc is now a detached system, the possibility of a
high value for mass transfer could be considered in the recent past,
when the system had an SD configuration. The existence of an extended circumbinary gas envelope in the system was pointed out
by Popper & Tomkin (1984) and Glazunova et al. (2008), while
photometric and spectroscopic evidence of a thin extended atmosphere around the secondary component was presented by Olson
& Etzel (1993). However, no emission lines from an accretion disc
around the primary have been found so far. The small mass ratio
(<0.1) of S Cnc suggests that the system is close to the end of
mass transfer in its nominal SD state (Popper & Tomkin 1984;
Olson & Etzel 1993). According to these results in the literature, the transferred mass rate could be taken at a low value of
10−10 M yr−1 for the system as per equation (16), and the computed lowest value of the mass-loss rate of 10−9 M yr−1 for S Cnc
among the other systems could be the cause of its decrease in orbital
period.
(i) Six Algols (FK Aql, AM Aur, WX Sgr, BS Sct, BD Vir,
XZ Vul) are neglected systems; their orbital period variations have
not been studied in detail so far.
(ii) The O−C diagrams of 10 Algols (AM Aur, RX Cas,
DK Peg, RV Per, WX Sgr, RZ Sct, BS Sct, W Ser, BD Vir, XZ
Vul) show upward parabolic forms indicating orbital period increase. Mass transfer was found to be the dominant mechanism for
observed orbital period changes in these 10 Algols. The average
mass transfer rate in the non-conservative cases for β 1 was calculated to be of the order of 10−7 M yr−1 , which is at the upper
limit of mass transfer rate generally accepted for Algol binaries
(e.g. 10−11 –10−7 M yr−1 from Richards & Albright 1999), except for W Ser and RX Cas, which have larger mass transfer rates
(≈10−5 M yr−1 ) than the other eight Algols.
(iii) For Algols with increasing orbital periods, as can be seen
in Fig. 19, the mass-loss parameter (β) tends to approach 1 (i.e.
β cri → 1) as |Ṁ 2 | decreases.
(iv) The O−C diagrams of eight Algols (FK Aql, S Cnc, RU Cnc,
TU Cnc, SX Cas, TW Cas, V548 Cyg, RY Gem) show downward
parabolic forms indicating orbital period decrease. Therefore, magnetic braking (or the Alfvén wave-driven stellar wind model) was
found to be the dominant mechanism for observed orbital period
decreases in these eight Algols. In this case, the average rate of
mass-loss due to stellar winds was calculated to be of the order of
10−8 M yr−1 for β βcri , which is higher than the upper limit
of typical rates of mass-loss due to wind from red giant stars (e.g.
10−11 –10−8 M yr−1 from Hilditch 2001).
(v) For Algols with decreasing orbital periods, as can be seen in
Fig. 20, the mass-loss parameter (β) tends to approach its critical
value as |Ṁ 2 | increases.
(vi) Three SD binary systems (W Ser, RX Cas, SX Cas) exhibit
the most interesting features among the 18 Algols studied here.
RX Cas and SX Cas are characterized as W Ser-type interacting
binary systems, while W Ser itself is a prototype of a highly interacting eclipsing binary system. In general, W Ser-type systems
consist of a mass-gaining primary component (a hot B, or late A,
normal star) embedded in an optically opaque thick accretion disc;
the mass-losing secondary component is a giant star of late spectral
type. W Ser and RX Cas show orbital period increase while SX Cas
displays orbital period decrease. The reason could be that since the
mass-gaining primary component of SX Cas is less obscured by surrounding circumstellar matter than that of W Ser (see Piirola et al.
2006), the angular momentum loss due to Alfvén wave-driven stellar wind from the late-type giant star of SX Cas is a more dominant
mechanism than mass transfer to its primary companion.
(vii) A linear relation between the orbital period (logP in units
of seconds) and its rate of change (log|P| in units of s yr−1 ) for
these 18 Algols was found:
log|P | = 1.88(95) × logP − 2.08(9).
(20)
This relation is shown in Fig. 21 (solid line) and in agreement
with the formula given by Prikhod’ko (1961), Oh (1991) and Qian
(2000a). As can be seen in Fig. 21, the result shows that the shorter
the orbital period, the lower the rate of change in the orbital period.
MNRAS 441, 1166–1176 (2014)
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A. Erdem and O. Öztürk
Figure 21. Empirical linear relation between orbital period (P) and its rate
of change (dP/dt) for 18 Algol-type binaries.
AC K N OW L E D G E M E N T S
This work was supported by TUBITAK (Scientific and Technological Research Council of Turkey) under grant no. 108T714. We
thank Mr G.H. Lee for checking the English.
REFERENCES
Andersen J., Nordstrom B., Mayor M., Polidan R. S., 1988, A&A, 207, 37
Brancewicz H. K., Dworak T. Z., 1980, Acta Astron., 30, 501
Budding E., 1984, Bull. Inf. Cent. Donnees Stellaires, 27, 91
Budding E., Erdem A., Çiçek C., Bulut I., Soydugan F., Soydugan E., Bakış
V., Demircan O., 2004, A&A, 417, 263
Dervişoğlu A., Tout C. A., İbanoğlu C., 2010, MNRAS, 406, 1071
Deschamps R., Siess L., Davis P. J., Jorissen A., 2013, A&A, 557, 40
Erdem A., Budding E., Demircan O., Değirmenci Ö. L., Gülmen Ö., Sezer
C., 2005, Astron. Nachr., 326, 332
Erdem A., Doğru S. S., Soydugan F., Çiçek C., Demircan O., 2010, New
Astron., 15, 628
Etzel P. B., Olson E. C., 1985, AJ, 90, 504
Giuricin G., Mardirossian F., Mezzetti M., 1983, ApJS, 52, 35
Glazunova L. V., Yushchenko A. V., Tsymbal V. V., Mkrtichian D. E., Lee
J. J., Kang Y. W., Valyavin G. G., Lee B.-C., 2008, AJ, 136, 1736
Gurzadyan G. A., 1997, Ap&SS, 257, 223
Hall D. S., 1989, Space Sci. Rev., 50, 219
Hayasaka T., 2000, in Percy J. R., Wilson J. B., eds, ASP Conf. Ser. Vol.
220, Amateur – Professional Partnerships in Astronomy. Astron. Soc.
Pac., San Francisco, p. 367
Hilditch R. W., 2001, An Introduction to Close Binary Stars. Cambridge
Univ. Press, Cambridge, UK
İbanoğlu C., Soydugan F., Soydugan E., Dervişoğlu A., 2006, MNRAS,
373, 435
İbanoğlu C., Dervişoğlu A., Çakırlı Ö., Sipahi E., Ÿüce K., 2012, MNRAS,
419, 1472
Kaitchuck R. H., Honeycutt R. K., Schlegel E. M., 1985, PASP, 97, 1178
Kalv P., 1979, Tartu Astrofüüs. Obs. Teated, 58, 3
Kopal Z., 1955, Ann. Astrophys., 18, 379
Kreiner J. M., Kim C.-H., Nha Il-S., 2001, An Atlas of O-C Diagrams of
Eclipsing Binary Stars. Pedagogical Univ. Press, Cracow, available at:
http://www.as.up.krakow.pl/o-c/
MNRAS 441, 1166–1176 (2014)
Mardirossian F., Mezzetti M., Cester B., Giuricin G., Russo G., 1980,
A&AS, 39, 235
Narita E., Schroeder K.-P., Smith R. C., 2001, Observatory, 121, 308
Oh K.-D., 1991, in Ashley M. C. B., Couch W. J., Caswell J. L., Hunstead
R. W., Proust K. M., eds, Proc. 5th Asian-Pacific Regional Astronomy
Meeting, Vol. 9, Period Changes of Ten Eclipsing Variable Stars. Astron.
Soc. Australia, Sydney, p. 289
Olson E. C., Etzel P. B., 1993, AJ, 106, 1162
Olson E. C., Etzel P. B., 1994, AJ, 108, 262
Pavlovski K., Kriz S., 1985, Bull. Astron. Inst. Czech., 36, 153
Peters G. J., 2001, in Vanbeveren D., ed., Astrophysics and Space Science
Library, Vol. 264, The Influence of Binaries on Stellar Population Studies. Kluwer, Dordrecht, p. 79
Pfeiffer R. J., Koch R. H., 1973, Inf. Bull. Var. Stars, 780, 1
Pfeiffer R. J., Koch R. H., 1977, PASP, 89, 147
Piirola V., Berdyugin A., Mikkola S., Coyne G. V., 2005, ApJ, 632, 576
Piirola V., Berdyugin A., Coyne G. V., Efimov Yu. S., Shakhovskoy N. M.,
2006, A&A, 454, 277
Plavec M. J., 1980, in Plavec M. J., Popper D. M., Ulrich R. K., eds, Proc.
IAU Symp. 88, Close Binary Stars: Observations and Interpretation.
Reidel, Dordrecht, p. 251
Plavec M. J., 1989, Space Sci. Rev., 50, 95
Plavec M. J., Weiland J. L., Koch R. H., 1982, ApJ, 256, 206
Popper D. M., Tomkin J., 1984, ApJ, 285, 208
Prikhodko A. E., 1961, Astron. Zh., 38, 937
Qian S., 2000a, AJ, 119, 901
Qian S., 2000b, AJ, 119, 3064
Qian S., 2001a, AJ, 121, 1614
Qian S., 2001b, AJ, 122, 1561
Qian S., 2001c, AJ, 122, 2686
Richards M. T., Albright G. E., 1999, ApJS, 123, 537
Sanad M. R., Bobrowsky M., 2013, Ap&SS, 344, 405
Sarma M. B. K., Vivekananda R. P., 1997, AJ, 114, 800
Sarna M. J., Muslimov A. G., Yerli S. K., 1997, MNRAS, 286, 209
Sarna M. J., Yerli S. K., Muslimov A. G., 1998, MNRAS, 297, 760
Soydugan F., Erdem A., Doğru S. S., Aliçavuş F., Soydugan E., Çiçek C.,
Demircan O., 2011, New. Astron., 16, 253
Svechnikov M. A., Kuznetsova E. F., 1990, Katalog Priblizhennykh Fotometricheskikh i Absoliutnykh Elementov Zatmennykh Peremennykh
Zvezd. Izd-vo UralÃskogo universiteta, Sverdlovsk
Todorova P. N., 1993, Inf. Bull. Var. Stars, 3904, 1
Todorova P. N., Khruzina T. S., 1989, Sov. Astron. Lett., 15, 225
Tout C. A., Hall D. S., 1991, MNRAS, 253, 9
van Hamme W., Wilson R. E., 1993, MNARS, 262, 220
van Rensbergen W., de Greve J. P., Mennekens N., Jansen K., de Loore C.,
2010, A&A, 510, 13
van Rensbergen W., de Greve J. P., Mennekens N., Jansen K., de Loore C.,
2011, A&A, 528, 16
Weiland J. L., Shore S. N., Beaver E. A., Lyons R. W., Rosenblatt E. I.,
1995, ApJ, 447, 401
Wilson R. E., 1989, Space Sci. Rev., 50, 191
Yang Y.-G., Wei J.-Y., 2009, AJ, 137, 226
Young A., Snyder J. A., 1982, ApJ, 262, 269
Zasche P., 2008, Doctoral thesis, Astron. Inst. Charles Univ., Prague
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