Download Chapter 2 Mass Transfer in Binary Systems

Chapter 2 Mass Transfer in Binary Systems
Chapter 2 Mass Transfer in Binary
Systems
1. Interacting binary systems and Roche lobe overflow
 Why binaries important?
(1) A majority of all stars are members of binary systems, which
may undergo mass transfer at some stage of their evolution
(Porb<103 days).
(2) Binary observations  astrophysical parameters  stellar
structure and evolution
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Chapter 2 Mass Transfer in Binary Systems
 Ways of mass transfer
(1) Roche lobe overflow caused by the nuclear expansion or the
shrinkage of the orbit due to loss of angular momentum.
(2) Accretion of stellar wind.
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Chapter 2 Mass Transfer in Binary Systems
 The Roche model
Consider the motion of a test particle in the gravitational potential
due to the binary components with mass M1 and M2.
Assumptions:
(1) Test particle.
(2) The two stars are spherically symmetric, and can be regarded as
point masses dynamically (mass concentrated for most stars).
(3) Circular orbit (tidal effect)
(4) Synchronization (tidal effect)
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Chapter 2 Mass Transfer in Binary Systems
In the frame of reference corotating with the binary system
centered on the center of mass, the Euler equation for any gas flow
between the two stars is


  1
  1
v
 (v  )v  f  P   R  2  v  P
t


where P is the pressure inside the star, −2𝜔
⃑ × 𝑣 is the Coriolis
force experienced by a non-stationary particle, and  is the angular
velocity of the binary,
2
GM 1 / 2

( 3 )
Porb
a
where M = M1 + M2 is the total mass of the binary, a is the binary
separation.
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Chapter 2 Mass Transfer in Binary Systems
The Roche potential, including ONLY the effects of both the
gravitational and centrifugal forces, is
GM 1 GM 2 1   2
(r )       (  r )
| s1 | | s 2 | 2
where
     
s1  r  r1 , s2  r  r2 .
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Chapter 2 Mass Transfer in Binary Systems
The characteristics of the Roche equipotentialsR(r) = const
(1) The equipotentials are also surfaces of equal pressure and
density.
(2) a  scale, q  shape.
(3) When r >> a, circular concentered at the center of mass;
when r →a, circular around the center of each of the two stars.
(4) There is a critical surface surrounding two stars but joining at
the inner Lagrange point L1, which is a saddle (unstable) point of
R (i.e. R(L1)=0). The confined part surrounding each star is
called its Roche lobe (RL).
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Chapter 2 Mass Transfer in Binary Systems
Sections in the orbital plane of
the Roche equipotentials.
Variation of the Roche potential
as a function of position on the
line connecting the two stars.
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Chapter 2 Mass Transfer in Binary Systems
When one of the stars fills its Roche lobe, any perturbation (e.g. in
the pressure) will push its envelope close to the inner Lagrange point
L1 over the L1 point into the Roche lobe of the companion star.
The Roche lobe can be regarded as the maximum size of the star.
Seidov (2004, ApJ, 603, 283) derived exact analytical formulas for
the potential and mass ratio as a function of Lagrangian point
position, in the classical Roche model of the close binary stars.
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Chapter 2 Mass Transfer in Binary Systems

Effect of radiation pressure
i. Distortion of the RL
ii. Changes in the Roche potentials
Drechsel et al. 1995, A&A, 294, 723
Dermine et al. 2009, A&A, 507, 891
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Chapter 2 Mass Transfer in Binary Systems
 Kopal’s classification of binary stars
(1) Detached binary,
(2) Semi-detached binary,
(3) Contact binary.
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Chapter 2 Mass Transfer in Binary Systems
2. How to measure the Roche lobe?
Assume that star 2 is the lobe filling star, mass ratio q =M2/M1.
The RL radius: the radius of a sphere having the same volume as the
lobe.
RL 2
0.49q 2 / 3

, 0q
2/ 3
1/ 3
a
0.6q  ln( 1  q )
0.44q 0.33

,
0.2
(1  q )
0.1  q  10
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（Eggleton 1983）
Chapter 2 Mass Transfer in Binary Systems
RL 2
2
q 1/ 3
q 1/ 3
 4/3 (
)  0.462(
)
a
3 1 q
1 q ,
0.1  q  0.8
(Paczynski 1971)
The distance b1 of the L1 point from the center of star 1 is
x1  b1 / a  0.5  0.227 log q
or
(1  x1 ) 3 (1  x1 12 )
q 3
x1 (3  3x1  x12 ) .
Note that RL1 and b2 can be obtained with the same formulae by
replacing q by q-1.
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Chapter 2 Mass Transfer in Binary Systems
RL 2
M 2 0.46
(
)
,
RL1
M1
0.1  q 10
An important quantity is the rate at which the RL radius responds to
the donor mass at constant M (total mass) and J (orbital angular
momentum),
d log RL
d log RL / a d log a
L 
 ( 1  q )(

)
d log M 2
d log q
d log q
 2.13q  1.67,
0  q  50
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Chapter 2 Mass Transfer in Binary Systems
Example: Application to lower main sequence stars
If 0.1  q  0.8, the mean density of the lobe-filling star is
M2
2
3
2 

110
P
gcm
hr
4 R23 / 3
For lower main sequence stars, the mass-radius relation is
m2 = M2/M⊙  R2/R⊙
so
M2
1.4
3
2 

gcm
4 R23 / 3 m22
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Chapter 2 Mass Transfer in Binary Systems
This gives a period-mass relation
M2/M⊙  0.11Phr
and period-radius relation
R2/R⊙  0.11Phr
More generally, there is a relation between the mean density of the
RL-filling star and a critical orbital period,
R23 2 0.2
Pcr  0.35 (day)
(
)
M 2 1 q
When mass transfer occurs, the binary separation and lobe size will
change gradually, the star will have to adjust its structure to
maintain lobe filling.
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Chapter 2 Mass Transfer in Binary Systems
3. Mass transfer and orbital evolution
If we ignore the spin angular momentums of the two stars, the total
(orbital) angular momentum is given by
J  (M 1a12  M 2 a22 )  a 2   GMa
where a1=(M2/M)a, a2=(M1/M)a, and =(M1M2)/M is the reduced
mass.
Logarithmically differentiating the equation with time gives
a 2 J 2M 2 2M 1 M




a J
M2
M1 M
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Chapter 2 Mass Transfer in Binary Systems
(1) Conservative mass transfer
M  0 , J  0 , M 1  M 2
We have
af
M1i M 2i 2 Pf
M 1i M 2i 3
2M 2
a
(
)
(
)

(1  q) 
ai
M1f M 2f , Pi
M 1f M 2f
a
M2
If q < 1, 𝑎̇ > 0, mass transfer from the less massive star to the more
massive one will cause the orbit to enlarge.
If q > 1, 𝑎̇ < 0, mass transfer from the more massive star to the less
massive one will cause the orbit to shrink.
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Chapter 2 Mass Transfer in Binary Systems
(2) Non-conservative mass transfer
M  0, J  0
First consider the effect of mass loss.
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Chapter 2 Mass Transfer in Binary Systems
Assume
(i) a fraction  of the mass lost from the star 2 is transferred onto
the star 1, the remaining part () is lost from the binary with the
specific angular momentum (𝑎22 𝜔) of the mass losing star 2;
(ii) a fraction of the transferred mass is captured by star 1, the
remaining part () is lost from the binary with the specific angular
momentum (𝑎12 𝜔) of star 1.
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Chapter 2 Mass Transfer in Binary Systems
i.e.,
M1  ( )M 2 ,
J  (1   )M 2 a22   (1   )M 2 a12
or
M 2 1     (1   )q 2
J M 2 1
q2

[
  (1   )
]
[
]
J M 2 1 q
1 q M 2
1 q
a 2 J 2M 2 2M 1 M




a
J
M2
M1 M
2M 2
q

[ (1  q) 
(1   )]
M2
2(1  q)
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Chapter 2 Mass Transfer in Binary Systems
Note that the binary evolution may be non-conservative even when
.
This happens when the orbital angular momentum is transferred into
rotational angular momentum by tidal interaction.
For example, the gravitational radiation transfers
momentum at a rate (Landau and Lifschitz 1951)
J
32 G 3
 4 1
|GR  
M
M
Ma
s
1
2
5
.
J
5 c
21
angular
Chapter 2 Mass Transfer in Binary Systems
The cause of the loss of angular momentum includes
mass loss,
gravitational radiation,
magnetic braking
J MB  M 2 RA2
The orbital evolution is
determined by the combined
effects of mass loss and
angular momentum loss.
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Chapter 2 Mass Transfer in Binary Systems
4. Stability of mass transfer and different mass transfer
timescales
When RL overflow occurs, the binary separation and lobe size will
change gradually, the star will have to adjust its structure to
maintain lobe filling for stable mass transfer
R2  RL , R2  RL , ( R  R / t )
or
R2 R L

R2 R L
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Chapter 2 Mass Transfer in Binary Systems
The exponents in power-law fits of radius to mass, R ~M, orζ
dlnR/dlnM, corresponding to
𝜕𝑙𝑛𝑅𝐿
| ,
𝜕𝑙𝑛𝑀2 bin
𝜁L =
𝜁ad =
loss,
ζth =
𝜕𝑙𝑛𝑅2
| ,
𝜕𝑙𝑛𝑀2 ad
∂𝑙𝑛𝑅2
| ,
∂𝑙𝑛𝑀2 th
the response of the RL to mass loss,
the adiabatic hydrostatic stellar response to mass
the thermal-equilibrium stellar response to mass loss
(generally close to zero).
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Chapter 2 Mass Transfer in Binary Systems
The timescale of mass transfer depends on the status of the outer
layers of the mass losing star.
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Chapter 2 Mass Transfer in Binary Systems
Stability requires that after mass loss (M2 <0), the star is still
contained by its Roche lobe. Assuming R2 = R2-R2L prior to mass loss,
the stability condition then becomes R < 0, or L< (ad, th).
(1) Nuclear timescale
The mass transfer is completely driven by slow expansion of the star
due to nuclear evolution, or by the contraction of its Roche lobe
caused by the loss of angular momentum by gravitational radiation
and/or magnetic stellar wind, etc.
The radius of the mass-losing star is identical to the Roche lobe
radius, and the star remains in thermal equilibrium.
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Chapter 2 Mass Transfer in Binary Systems
(2) Dynamical timescale
If L >ad, the lobe-filling star loses mass, but cannot remain within
its Roche lobe with hydrostatic equilibrium, even with extremely
rapid mass loss.
For stars with deep surface convective zones (lying on or near the
giant branch, or on the lower main sequence), and degenerate stars,
the variation of the radius of a star with deep convective envelope
depends on its mass as R M.
If mass is lost, the star expands, and cannot maintain within its
Roche lobe, leading to very rapid mass transfer and the formation of
common envelope.
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Chapter 2 Mass Transfer in Binary Systems
(3) Thermal timescale
If ad > L > th, the lobe filling star loses mass, and may remain within
its Roche lobe in hydrostatic, but not thermal equilibrium. This
applies to stars with radiative envelope.
The deviation from thermal equilibrium allows the star to remain
just filling the Roche lobe, and relaxation toward thermal
equilibrium drives mass on thermal timescale.
Thermal timescale mass transfer usually occurs when the more
massive star begins to fill its Roche lobe and transfer mass to its
companion, at a rate of
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Chapter 2 Mass Transfer in Binary Systems
𝑀2
̇
𝑀2 ≈ − ,
𝑡KH
where the Kelvin-Helmlholtz timescale
𝑡KH =
𝐺𝑀2
𝑅𝐿
≈ 3 × 107 (
𝑀 2 𝑅 −1 𝐿 −1
) ( ) ( )
𝑀⨀
𝑅⨀
𝐿⨀
29
yr
Chapter 2 Mass Transfer in Binary Systems
Example: Supersoft X-ray sources
Supersoft X-ray sources were first discovered with Einstein
Observatory and about four dozen new with the ROSAT satellite.
http://lheawww.gsfc.nasa.gov/users/white/wgacat/apjl.html
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Chapter 2 Mass Transfer in Binary Systems
These sources with supersoft X-ray radiation (emission dominantly
below 0.5 keV which corresponds to effective temperatures 50
eV),and bolometric luminosities in the range  erg/s.
It is widely believed that most of the
supersoft
X-ray
sources
are
accreting white dwarfs in binary
systems with a more massive donor,
and mass transfer occurs on thermal
timescales, so that there is stable
nuclear burning on top of the white
dwarfs.
Fujimoto 1982
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Chapter 2 Mass Transfer in Binary Systems
5. Different cases of mass transfer
Kippenhahn and Weigert defined three types of close binary
evolution.
Case A: mass transfer occurs when
the star is filling its RL before
attaining the firs relative minimum
of the radius, hence during the
core hydrogen burning (main
sequence) phase.
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Chapter 2 Mass Transfer in Binary Systems
Case B: mass transfer occurs after the end of core hydrogen
burning but before helium ignition inside the lobe-filling star (shell
hydrogen burning).
Case C: mass transfer occurs during core helium burning in the lobe
filling star.
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Chapter 2 Mass Transfer in Binary Systems
6. Spherically symmetrical accretion
Application:
(1) isolated neutron stars/black holes accreting from interstellar
medium;
(2) neutron stars/black holes accreting from the wind from their
massive, early type companion star in close orbits.
First consider a compact star at rest in interstellar medium. We
take spherical coordinate (r, ,) with origin at the center of the
star. The spherical symmetry implies that all the variables are
independent of  and , and have only the radial component.
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Chapter 2 Mass Transfer in Binary Systems
The mass continuity equation for steady
flow is

   (v )  0
t
1 d 2
 2
(r  v)  0
r dr
Note that here v < 0 for accretion flow (v > 0 for a stellar wind). The
integration gives the accretion rate
 4r 2 v  M
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Chapter 2 Mass Transfer in Binary Systems
The Euler equation
v
1
GM 1
 (v  )v  f  P   2  P
t

r

v
dv 1 dP GM

 2 0
dr  dr
r
We introduce the polytropic relation
P K
where K is a constant, and 1 (isothermal)  5/3 (adiabatic)
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Chapter 2 Mass Transfer in Binary Systems
Since
1 dP 1 dP d cs2 d


 dr  d dr  dr ,
and
1 d
1 d
  2 (vr2 )
 dr
vr dr
dv 1 dP GM
v 
 2 0
dr  dr
r
dv cs2 d
GM
2
v  2
(vr )  2  0
dr vr dr
r
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Chapter 2 Mass Transfer in Binary Systems
Before integrating the above equation, we may rewrite it as
cs2 d 2
2GM
2rcs2
2GM
r
(1  2 ) (v )   2 (1 
)   2 (1  )
v dr
r
GM
r
rs
where
rs  (1/ 2)GM / cs2 (rs ) .
It can be seen that the right hand side of the equation is equal to
zero at rs. This means that at rs either
v 2  cs2
or dv 2/dr0.
The latter case is irrelevant here.
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Chapter 2 Mass Transfer in Binary Systems
So we call rs the sonic point, and the accretion flow is trans-sonic,
i.e., v > cs when r < rs, and v < cs when r > rs.
Now integrating the Euler equation gives
v2
cs2
GM


 constant
2  1
r
The constant must be
cs2 () /(  1)
as r .
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Chapter 2 Mass Transfer in Binary Systems
When rrs,
v 2  cs2
so
2GM
v 
 v 2ff
r
2
At rrs,
v 2  cs2
so
2
1
1
c
2
s ( )
cs (rs )( 
 2) 
2  1
 1 ,
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Chapter 2 Mass Transfer in Binary Systems
or
cs (rs )  cs ()(
2 1/ 2
)
5  3
cs2 ~ P /  ~   1
cs (rs ) 2 /( 1)
2 1/( 1)
  (rs )   ()[
]
  ()(
)
cs ()
5  3
The mass accretion rate is
M  4r 2  (v)  4rs2  (rs )cs (rs )   (GM ) 2  (rs )cs3 (rs )
  (GM ) 2  ()cs ()  3 (
2 (5  3 ) / 2( 1)
)
5  3
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Chapter 2 Mass Transfer in Binary Systems
For interstellar medium with number density ~ 1 particles cm -3 and
temperature ~ 10 K (cs()~10 kms-1), and, the mass accretion
rate for a 1 M⊙ neutron star is around 1011 gs-1, or the accretion
luminosity ~1031 ergs-1.
The accretion rate can also be expressed as
2
M  racc
(  )cs (  )
where
racc
2GM
 2
cs ()
is called the accretion radius.
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Chapter 2 Mass Transfer in Binary Systems
For r > racc, the thermal energy cs(r)2/2 of a gas element is larger
than the gravitational binding energy GM/r.
The accretion radius gives the range of influence of the star on the
gas cloud. At this point (r) and cs(r) begin to increase above their
ambient values.
When the star is moving with a
velocity v in the gas cloud, the
accretion radius is
racc
2GM
 2
cs ()  v 2 .
Toropina et al. 2012, MNRAS, 420, 810
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Chapter 2 Mass Transfer in Binary Systems
When the star is orbiting a massive, early type companion star in a
close binary, the accretion radius is
2GM
2GM
racc  2  2
2 ,
vrel
vw  vorb
where vw~103 kms-1 is the wind velocity,
and the mass loss rate is
 M w  4a2 w ( a )vw ( a ) .
The mass accretion rate is
2
𝑀̇ = 𝜋𝑟acc
𝜌w 𝑣rel
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Chapter 2 Mass Transfer in Binary Systems
The fraction of the stellar wind captured by the compact star is
given by the mass flux into the accretion cylinder of radius racc,
2
𝑀̇
𝜋𝑟acc
𝑣rel
(𝐺𝑀)2
≈
= 2
3
2
̇
−𝑀𝑤 4𝜋𝑎 𝑣w 𝑎 𝑣w 𝑣rel
which is of order 10-4-10-3 for typical parameters of high-mass
X-ray binaries.
Given the mass loss rate of early type stars is ~10-6-10-5 M⊙yr-1, the
accretion luminosity can be as high as 1036-1038 ergs-1 depending on
the orbital periods.
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Chapter 2 Mass Transfer in Binary Systems
References
1. Frank, J., King, A. and Raine, D. 2002, Accretion power in
astrophysics, chapters 1 and 4.
2. Soberman, G. E., Phinney, E. S. and van den Heuvel, E. P. J. 1997,
A&A, 327, 620
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