I'll review a few aspects of a class of binary stars in which the secondary- the donor,
is a low mass star which is still on the main sequence or about to drift away from the
main sequence. The primary is a highly evolved star, mostly a white dwarf (CV's) or a
Neutron Star (LMXB's). The secondary
fills up the Roche lobe resulting in mass transfer to the compact primary thru a stream
and accretion disk.
The formation of binary stars in general is an open subject and not very well understood.
So we'd consider only the evolution of CV's once they are somehow formed. We shall see that
the evolution is primarily dictated by angular momentum loss mechanisms and the evolution
of the secondary star, as it tries to adjust its radius to the mass loss (due to mass
transfer and/or winds). Thus the relevant time-scales are angular momentum loss time-scales
, Kevin-Helmholtz time-scales and also mass loss time-scales.
Though potentially relevant, the details of Accretion disk formation and/or transport
of angular momentum in it wont be considered.
Cataclysmic Variables (CV's) are a class of short-period binary systems which have a low mass
secondary transferring mass onto a White Dwarf (WD) primary via Roche lobe overflow. The
evolution of the CV is governed by angular momentum losses, and the dynamical responses
of the binary system and the secondary to the mass transfer. The angular momentum loss is
dominated by magnetic braking (at longer periods) and gravitational radiation (at shorter
periods). Though the periods of CV's extend from approx. 1.3 hrs to 10 hrs, empirically it
is seen that there is a distinct dearth of such systems between 2 and 3 hrs. One of
the aims of any theory about CV's is to explain this 'period gap'. Another question that
needs to be addressed is whether systems evolves across this period gap or the two populations
above and below the gap are of different origin.
In general, a given CV will evolve from a higher mass secondary with a long
period to a lower mass secondary, short period system. Presently, it is believed that they
evolve through the period gap, becoming 'invisible' in the gap, and re-appearing at the other
end of the gap. This is achieved by modelling the secondary star is such a way as to be
consistent with the above requirements. From time to time however, it has been speculated
that the CV's on either side of the period gap may be entirely different populations.
In the following section, i'll outline the basic physics involved in the problem by setting
up the equations for mass transfer and angular momentum loss. Then i intend to
work out in detail the equations in case of non-conservative mass transfer, i.e. the effect
of winds and inefficient mass transfer. I then intend to work on the possibility of a
'Circumbinary Disk' and its influence on the binary.
Finally, i'll try and outline what work can be done in this field which would
be interesting, relevant and manageable!
We start with the orbital angular momentum equation from Kepler's law:
J = M1 M2
æ è
G a
M
ö ø
1/2
(1)
To study the time evolution, we log differentiate
×
J
J
=
×
M1
M1
+
×
M2
M2
+
1
2
×
a
a
-
1
2
×
M
M
(2)
Here,
M1 is the Mass of the primary(accretor),
M2 is the Mass of the secondary(donor),
M = M1 + M2 is the total Mass of the system and,
a is the binary separation.
(2) gives the functional dependance of the rate at which
angular momentum is lost from the system in terms of the Masses
involved and the separation. However, one can write the angular
momentum loss rate in terms of the various 'modes' in which it
can be lost as :
d J
d t
=
æ è
¶J
¶t
ö ø
GR
+
æ è
¶J
¶t
ö ø
[M\dot]
(3)
The first term GR, represents the angular momentum loss rate as
a result of gravitational radiation1.
[M\dot] represents the total mass loss rate from the
system. This may be in the form of an isotropic wind from the
secondary (w2), or wind from a 'circumstellar' disk or
ring as matter accretes onto the primary (w1), or may also be due to
mass loss from a 'circumbinary' ring (cb) [1]. Thus, i can write :
These considerations are important because if the mass loss in any
of these modes is significant, the overall evolution of the system
will be correspondingly affected - the specific angular momentum
lost by the system depends on which of these modes dominates.
Each of these modes can be parameterized by associating a
dimensionless specific angular momentum parameter with it.
For example, in the case of 'Conservative' mass transfer,
all these parameters are set to zero. Let us specialize in the
'Conservative' mass transfer case, and later we can generalize to
the non-conservative case.
The equations governing the evolution of CV's is simplified
greatly in this case, since we can set :
×
M
= 0,
×
M
1
= -
×
M
2
(6)
Consequently, (2) implies that the separation of the binary
shrinks as the angular momentum is lost via Gravitational
radiation, since the second term in (3) is now 0. The rate
at which angular momentum is lost by this mechanism is
calculated [2], and is given
in the weak field limit by :
æ è
×
J
J
ö ø
GR
= -
32
5
G3
c5
M1 M2 M
a4
(7)
To estimate in what regimes this mechanism is most efficient, we
recall the emperical/numerical relation :
RL
a
=
2
34/3
æ è
q
1+q
ö ø
1/3
= 0.462
æ è
M2
M
ö ø
1/3
(8)
Here,
RL is the Roche-lobe Radius,
q = M2/M1 is the mass ratio.
Using (8) to eliminate a in (7) and stable mass transfer, i.e. RL = R2, we get
æ è
×
J
J
ö ø
GR
µ
M1
M1/3
M27/3
R24
(9)
Now, if the secondary is near the main sequence, R2 µ M2, and so, ([J\dot]/J)GR ~ M2-5/3 µ P-5/3.
Also, log differentiating (8), using (2) , we get :
Equation (12) gives the estimate of mass transfer rates driven by
Gravitational Radiation. This mass transfer rate, in turn,
dictates how luminous the system is. The Accretion Luminosity is
given by [4] :
Lacc =
G M1(-
×
M
2
)
R1
(13)
Though a direct measurement of -[M\dot]2 is not possible to fix
Lacc, (12) implies that for periods ~ 2 hrs., -[M\dot]2 is consistent with expected Lacc values( ~ 1032 erg/s) for M1 ~ 1 M\odot, R1 ~ 109 cm for CV's.
However, the mass transfer rates for periods > 2 hrs, arent
sufficient enough to account for the expected luminosities.
Moreover, 12 implies that higher period binaries would have
lower mass transfer rates and thus lower luminosity. This is
contradictory to what is observed. Longer period binaries are
significantly more luminous than shorter period ones.
Thus we can conclude that at shorter periods ( £ 2hrs)
angular momentum losses by Gravitational radiation are sufficient
to drive mass transfer rates which can account for the expected
luminosities. For longer periods though, we need some other
mechanism for angular momentum loss, which can drive the mass
transfer rates to higher levels, and in turn the luminosity. That
mechanism is magnetic stellar wind braking, or simply
magnetic braking.
2.2 Stellar Winds and Non-Conservative Mass Transfer
We've seen that angular momentum loss due to gravitational
radiation is not sufficient to explain the behavior of CV's above
P ³ 2 hrs. In fact, as mentioned earlier we dont see many
CV's in the 2-3 hr range. Above 3 hrs, the mass transfer
rates have to be much more than what GR can account for to
explain the expected luminosities. So we turn to the 2nd
mode for angular momentum loss in (3), mass loss3, to dictate the
evolution of the CV4.
Now, in the absence of a magnetic field we can ignore the spin (see below) on
the stars and concentrate on the orbital angular momentum of the
stars. Also lets set the mass lost due to winds by the primary, Mw1, to
0. Thus, we have a CV transferring mass from the secondary to
the primary, and the secondary is also loosing mass via a stellar
wind. The angular momentum lost due to this wind is
characteristic of the material associated with the secondary.
Also since the mass transfer is no longer conservative, the
assumptions (6) no longer hold. Instead, we have ...
×
M
1t
= -b
×
M
2t
,
(14)
×
M
=
×
M
2w
+
×
M
1t
+
×
M
2t
=
×
M
2w
+ (1-b)
×
M
2t
The net angular momentum loss from the system in such a situation
can be given as ...
d J
d t
=
æ è
¶J
¶t
ö ø
GR
+
æ è
¶J
¶t
ö ø
[M\dot]
+
æ è
¶J
¶t
ö ø
[M\dot]t
(15)
Here, the last term now represents the angular momentum loss due
to mass transfer in the presence of a magnetic field. The twisting
of the field lines due to the rotating fluid in the disc
helps generate a torque on the secondary5.
Explicitly substituting in the various term and using equations
(14) into (15), we get
æ è
×
J
J
ö ø
=
1
J
é ë
æ è
¶J
¶t
ö ø
GR
+ aw
×
M
2w
Wa22 +aL(1-b)
×
M
2t
Wa12 + aD
×
M
2t
W(a-RL)2
ù û
(16)
Here,
the a's represent parameters that may depend on detailed
modelling of that particular loss mechanism, and the a's
represent the length scale or lever arms for that
mechanism. The subscripts w, L and D stand for 'wind',
'Loss' and 'Direct' respectively. Also,
J = m
Ö
G M a
,
m =
M1 M2
M
= M2
æ è
1
1+q
ö ø
.
Substituting these equations into (16) we get ...
æ è
×
J
J
ö ø
=
æ è
×
J
J
ö ø
GR
+
aw
×
M
2w
M2
(1+q)
æ è
a2
a1
ö ø
2
+
aL(1-b)
×
M
2t
M2
(1+q)+
aD
×
M
2t
M2
(1+q)
æ è
a-RL
a
ö ø
2
(17)
Now, the log derivative of (8) is
×
R
L
RL
=
×
a
a
-
1
3
×
M
M
+
1
3
×
M
2
M2
(18)
Solving (2) for [a\dot]/a, and substituting from (17) we get:
×
a
a
= 2
æ è
×
J
J
ö ø
GR
+ 2
×
M
2w
M2
é ë
aw (1+q)
æ è
a2
a1
ö ø
2
- 1 +
1
2
q
1+q
ù û
+
2
×
M
2t
M2
é ë
aL(1-b)(1+q)+ q b- 1+
1
2
q
1+q
(1-b) + aD (1+q)
æ è
a-RL
a
ö ø
2
ù û
(19)
Now substituting (19) into (18)
×
R
L
RL
= 2
æ è
×
J
J
ö ø
GR
+
×
M
2w
M2
é ë
aw (1+q)
æ è
a2
a1
ö ø
2
- 1 +
1
2
q
1+q
+
1
6(1+q)
ù û
+ 2
×
M
2t
M2
é ë
aL(1-b)(1+q)+ q b- 1 +
1
2
q
1+q
(1-b) + aD (1+q)
æ è
a-RL
a
ö ø
2
+
b
6
q
1+q
+
1
6
1
1+q
ù û
(20)
Now, the secondary is a main sequence star and so we can exploit
the mass-radius relation for MS stars along with the contact
condition which we assume holds for our binary system, so that
...
×
R
2
R2
= xMS
×
M
2
M2
Þ
×
R
L
RL
= xMS
×
M
2
M2
= xMS
é ë
×
M
2w
M2
+
×
M
2t
M2
ù û
(21)
Now, substitute (21) on the LHS of (20) and rearrange to get:
×
M
2t
M2
=
2
æ è
×
J
J
ö ø
GR
-
×
M
2w
M2
é ë
xMS - {aw (1+q)
æ è
a2
a1
ö ø
2
- 1 +
q
2(1+q)
+
1
6(1+q)
}
ù û
2
é ë
xMS-{aL(1-b)(1+q)+ q b- 1 +
q(1-b)
2(1+q)
+ aD (1+q)
æ è
a-RL
a
ö ø
2
+
bq +1
6(1+q)
}
ù û
(22)
This then, is a most general description of the mass transfer rate
as dictated by various angular momentum loss mechanisms,
discounting the possible affects of a 'Circumbinary disk' which
will be considered next.
· Reality Check: To get to (11), which is the simplified, conservative
counterpart of (22), we set:
[M\dot]2w = 0, xMS = 1, b = 1, aw = 1, aD = 06. On rearranging, we indeed get back (11)!
The circumbinary disk, can in principle extract infinite amount of
angular momentum from the system [5]. The key is to able to
cut off this mechanism, along with the other mechanisms, at the
upper end of the period gap. This prolly implies magnetic coupling
between the disk and the binary. Also, the disk has to be
maintained for it to be relevant for long term evolution. Thus the
disk may be static(extending outward to infinity?) or may be fed
by mass flowing out of the binary [6]. On the other hand it
should'nt be too bright, so as to be observable easily (since we
dont seem to see any such disk's around binaries too often!). The
angular momentum transport within this disk thus also needs to be
investigated [7].
I am working on all this....
Initially thought i'd look at what people have done here, but the
circumbinary disk is more interesting. Important thing to remember
here is the very short time scales at which a lot of mass can be
lost from the system and so also angular momentum. The system may
react violently, even be disrupted if the secondary is burping
from time to time [8]!
I need to generalize (22) further to include the circumbinary
disk. In spite of not doing so (and its apparent ugliness!),
(22) is the most general form of the mass transfer rate in terms
of various angular momentum loss mechanisms. Detailed modelling
would be required to set the various parameters, like the
a's and the lever arms (the a's).
1GR experts, please opine!
2Why Partials?
3i treat
'winds' as essentially a loss of mass from the system
4the circumbinary ring is too hot to handle!
lets leave that for later.
5Why isnt this
term present in the conservative case?
6Actually,
this term should've been included in the Conservative case, but since i didnt,
i'am setting it to zero.