Chapter 4 Formation and Evolution of X-ray Binaries
Chapter 4 Formation and Evolution of
X-ray Binaries
1. Prerequisites for NS/BH formation
NS/BHs are end products of the
evolution of massive stars.
Stars on the supergiant branch are very
large with radii of up to ~103 R⊙ that is
usually much larger than the current
orbits of XRB
1
Chapter 4 Formation and Evolution of X-ray Binaries
2/3
a  2.9 1011 m11/ 3 (1  q)1/ 3 Pday
cm
Let us take typical parameters for an LMXB, say MNS ≈ 1.4 M⊙,
M2 ≤1 M⊙, and orbital separation a ~ R⊙, the orbital angular
momentum is
The progenitor of the NS should be more massive than ~ 10
M⊙ with maximum radius ~102-103 R⊙. Therefore, the initial
orbital angular momentum of the binary is
2
Chapter 4 Formation and Evolution of X-ray Binaries
3
Chapter 4 Formation and Evolution of X-ray Binaries
Comparing now the total mass and orbital angular momentum of
an LMXB with the corresponding values of its progenitor
system we find that Mi /MLMXB ~10-30 and Jorb,i / JLMXB ~ 102.
In other words: the formation of an LMXB invokes a binary
evolution in which the progenitor system has to lose ~90% of
its initial mass and up to ~99% of its initial orbital angular
momentum.
4
Chapter 4 Formation and Evolution of X-ray Binaries
2. Common envelope (CE) evolution
The CE evolution occurs when either of the following
conditions is fulfilled:
(1) the mass ratio q = M2/M1 is larger than a critical value;
(2) the envelope of the donor star is in convective equilibrium.
HMXBs usually have very large mass ratios and are typical
examples of systems that are bound to undergo CE evolution.
5
Chapter 4 Formation and Evolution of X-ray Binaries
(3) the Darwin instability
Tidal friction attempts to dissipate relative motion and lead to
a state of uniform rotation, with the components corotating
with the binary in circular orbits.
Consider a detached binary with a circular orbit, where star 1
is sufficiently small that its moment of inertia can be
neglected. The total angular momentum, if star 2 corotates, is
2/3
(
GM
)
M2
2
J  M 2R  1/ 3
 (1  q)
where R is the radius of gyration of star 2.
6
Chapter 4 Formation and Evolution of X-ray Binaries
Through evolution R may grow with time. However, the above
equation when differentiated wrt  shows that R, considered
as function of , has a maximum value, say R0, which occurs
when
1 (GM ) 2 / 3 M 2
2
M 2R0 
3 1/ 3 (1  q)
i.e. when the ratio of spin to orbital angular momentum is 1/3.
If R grows beyond R0, corotation must break down. The star
takes angular momentum from the orbit, which shrinks and
rotates faster.
7
Chapter 4 Formation and Evolution of X-ray Binaries
In all these cases, the mass transfer rates are so high that
only a small fraction of the transferred mass can be accreted.
The excess material is expelled to surround the binary stars,
producing a common envelope.
The secondary star, while moving through the envelope of its
companion, experiences a very large frictional drag, which
causes its orbit to shrink rapidly.
8
Chapter 4 Formation and Evolution of X-ray Binaries
A computer simulation of common
envelope evolution in which the two
stars, in this case a giant and a main
sequence star, are initially separated
by ~1 AU and evolve (in just under
1 year) until they are almost
touching. Note how the common
envelope is ejected at the end of the
process
(http://www.shef.ac.uk/~phys/people
/vdhillon/seminars/sas/ce.html).
9
Chapter 4 Formation and Evolution of X-ray Binaries
(From astro-ph/0611043)
10
Chapter 4 Formation and Evolution of X-ray Binaries
CE evolution leads to
(1) a very close binary system, consisting of the secondary star
and the compact core of the primary, if the orbital energy
deposited in the envelope is sufficient to blow off the
envelope.
(2) a Thorne-Zytkow object (TZO), and finally a single star,
due to coalescence of the secondary with the core of the
primary.
Evidence of CE evolution: The very short orbital periods of CVs and the binary radio
pulsars PSR 1913+16 and PSR 0655+64 indicate that these systems must be the
results of CE evolution.
11
Chapter 4 Formation and Evolution of X-ray Binaries
Assume that a fraction CE of the orbital energy,
 CE Eorb
GM 2c M1 GM 2 M1
  CE (

)
,
2af
2ai
is used to eject the envelope with binding energy
GM 2 M 2e GM 2 M 2e
Ebind 

R2
ai rL
where M2 = M2c (core) + M2e (envelope) is the mass of the donor
star, ai and af the orbital separation of the binary before and
after CE evolution, rL = R2L/ai the dimensionless Roche lobe
radius of the donor star at the onset of the mass transfer, and
λ a factor (<1) taking into account the mass distribution in the
12
Chapter 4 Formation and Evolution of X-ray Binaries
envelope.
From the equation CEEorb= Ebind, the ratio of final (post-CE) to
initial (pre-CE) orbital separation can be obtained as
af
M 2cM1
1
(
)[
]
ai
M2
M1  2M 2e /( CErL ) .
13
Chapter 4 Formation and Evolution of X-ray Binaries
The total binding energy of the envelope to the core is given by
M2
GM (r )
Ebind  
dm   th  Udm
M 2c
M 2c
r
M2
where the first and second terms are the gravitational binding
energy and the internal thermodynamic energy involving the
thermal energy of a perfect gas, the energy of radiation, as
well as terms due to ionization of atoms and dissociation of
molecules and the Fermi energy of a degenerate electron gas .
The value of αth (0~1) depends on the details of the ejection
process, which is very uncertain.
14
Chapter 4 Formation and Evolution of X-ray Binaries
Example 1
Consider the evolution of a binary that initially has component
masses of 15 M⊙ and 2 M⊙, evolving according to case B or case
C mass transfer.
The 15 M⊙ star at the end of core hydrogen burning has a
helium core of 4 M⊙, and a hydrogen rich mantle of 11 M⊙, so
M2c=4 M⊙, M2e=11 M⊙, and M1=2 M⊙.
Application of the above equation with λ = 0.5 and CE =1 then
yields ai/af =168, implying that in order to throw off the
envelope, the orbit should shrink by this factor.
15
Chapter 4 Formation and Evolution of X-ray Binaries
A binary system will survive only if after spiral-in the Roche
lobe radius a1rL1 ≥ R1.
For a 2 M⊙ star with a 4 M⊙ helium star companion one has rL1 =
0.32 and R1 ≥ 1.5R⊙, which implies af ≥ 4.5R⊙.
Consequently, ai should have been ≥756 R⊙, or 3.5 AU.
16
Chapter 4 Formation and Evolution of X-ray Binaries
Systems with the same component masses but smaller orbital
separations will not have survived spiral-in, but will have
coalesced completely.
For a 15 M⊙+1 M⊙ system the survival condition becomes ai/af
=330 and ai ≥ 6 AU.
Since a 15 M⊙ star does not reach a radius larger than 1000 R⊙
(4.5 AU) before reaching core collapse, it will never overflow
its Roche lobe before it explodes as a supernova.
17
Chapter 4 Formation and Evolution of X-ray Binaries
Example 2
Formation of compact BH LMXBs (Podsiadlowski et al. 2003,
MNRAS, 341, 385; Justham et al. 2006, MNRAS, 366, 1415)
<0.1 for BH progenitors (M>25M⊙)
18
Chapter 4 Formation and Evolution of X-ray Binaries
af
rL M 2c
 (
) M 1
ai
2 M 2 M 2e
R2,max
M1
 a f  0.011(ai rL ) M 1  (25R )(
)(
)
M  2300 R
2300 R M 
   0.15(
)
R2,max
M1
It is difficult to form compact
BH LMXBs with the standard
channel.
19
Chapter 4 Formation and Evolution of X-ray Binaries
3. Supernovae in close binaries
When one component of a binary explodes symmetrically as a
supernova (SN), its mass will suddenly decrease, which causes
a decrease in the binding energy of the system.
Consider a binary of stars
with masses M and m in a
circular orbit of radius ai,
and assume that the star
with mass M suddenly loses
an amount of mass ΔM due
20
Chapter 4 Formation and Evolution of X-ray Binaries
to a SN explosion.
The assumptions that the positions and velocities of both stars
are the same immediately before and after the explosion
implies that the velocities immediately after the explosion are
perpendicular to the line connecting the stars, i.e., that the
periastron distance of the new orbit is equal to the radius of
the old orbit:
ai = af (1-e)
where af and e are the semi-major axis and the eccentricity of
the new orbit.
21
Chapter 4 Formation and Evolution of X-ray Binaries
It further implies that the relative velocity at periastron
equals the relative velocity of the circular orbit:
G( M  m) G( M  m  M ) 1  e

ai
af
1 e
From these two equations one may derive the eccentricity of
the final orbit
M
e
m  M  M ,
the change of the semi-axis
af
m  M  M

ai m  M  2M ,
22
Chapter 4 Formation and Evolution of X-ray Binaries
and the velocity of the center of gravity of the post-explosion
orbit with respect to its velocity before the explosion
mv  ( M  M )V
vs 
 eV
m  M  M
where v and V are the velocities in the old, circular orbit of the
stars with masses m and M, respectively.
It can be seen from above equations that if more than half of
the system mass is ejected during the explosion, the system
will become unbound, and the stars will leave each other in
hyperbolic orbits with their orbital velocities.
23
Chapter 4 Formation and Evolution of X-ray Binaries
There is now firm evidence that most newborn neutron stars
receive a momentum kick at birth which gives rise to high
velocities (~400 kms-1).
For an asymmetric supernova, the change of the semi-major
axis is
af
1  M /( M  m)

ai 1  2M /( M  m)  ( w / vrel ) 2  2 cos  ( w / vrel )
where w is the magnitude of the kick velocity, vrel =
[G(M+m)/ai]1/2 is the relative velocity in pre-SN binary, and θ is
the direction of the kick velocity relative to the orientation of
the pre-SN velocity.
24
Chapter 4 Formation and Evolution of X-ray Binaries
For each binary there exists a critical angle θcrit for which a
supernova with θ < θcrit will result in disruption of the orbit.
25
Chapter 4 Formation and Evolution of X-ray Binaries
4. Formation and evolution of X-ray binaries
(1) End products of X-ray binaries – binary and millisecond
pulsars (BMSPs)
26
Chapter 4 Formation and Evolution of X-ray Binaries
More than 70 BMSPs known in the Galactic disk can be roughly
divided into four classes:
(i) High-mass, close-orbit (Porb≤15 days) BMSPs with a NS or a
relatively heavy CO/ONeMg WD companion.
(ii) Low-mass BMSPs with a helium WD companion (MWD < 0.45
M⊙).
(iii) Non-recycled pulsars with a CO WD companion.
(iv) Pulsars with an un-evolved companion.
Two “planet pulsars” PSRs B1257+12 and B1620-26.
27
Chapter 4 Formation and Evolution of X-ray Binaries
Main type
Sub-type
High-mass companion
(0.5≤MC/M⊙≤1.4)
NS + NS (double)
NS + (ONeMg) WD
NS + (CO) WD
Low-mass companion
(MC≤ 0.45M⊙)
NS + (He) WD
Non-recycled pulsar
(CO) WD + NS
Un-evolved companion
B-type companion
Low-mass MS
companion
28
Observational examples
PSR 1913+16,
Porb =7.75 hr;
PSR 1435-6100,
Porb =1.35 d;
PSR 2145-0750,
Porb =6.84 d;
PSR 0437-4715,
Porb =5.74 d
PSR 1640+2224,
Porb =175 d
PSR 2303+46,
Porb =12.3 d
PSR 1259-63,
Porb =3.4 yr
PSR 1820-11,
Porb =357 d
Chapter 4 Formation and Evolution of X-ray Binaries
29
Chapter 4 Formation and Evolution of X-ray Binaries
30
Chapter 4 Formation and Evolution of X-ray Binaries
Binary pulsars with
low-mass companions
( <0.7 M⊙ WDs) have
essentially circular orbits
(10-5 <e<0.01).
Binary pulsars with
high-mass companions (>
1M⊙ – massive WDs, other
NSs or main sequence stars)
tend to have more eccentric
orbits, 0.15<e< 0.9.
31
Chapter 4 Formation and Evolution of X-ray Binaries
32
Chapter 4 Formation and Evolution of X-ray Binaries
Double neutron star binaries
33
Chapter 4 Formation and Evolution of X-ray Binaries
34
Chapter 4 Formation and Evolution of X-ray Binaries
NS-WD binaries: LMBPs and IMBPs
35
Chapter 4 Formation and Evolution of X-ray Binaries
The recycling scenario
SN of the more
massive star in the
binary
NS (high field,
rapid spin)
accretion
of
mass
and
angular
momentum onto
Magnetic field decay,
spin accelerated
All the envelope
is transferred
BMSP
36
NS (high field,
slow spin)
Secondary star
evolves
and
mass transfer
occurs
Chapter 4 Formation and Evolution of X-ray Binaries
(2) Formation and evolution of HMXBs
37
Chapter 4 Formation and Evolution of X-ray Binaries
 Formation of double NS/BH binaries
All HMXBs end up in a common envelope phase, as the neutron
star (or low-mass black hole) is engulfed by the extended
envelope of its companion, in an orbit which is rapidly shrinking.
There are large uncertainties
about the wind mass losses,
helium star evolution and
supernova explosions during
the formation and evolution
of HMXBs.
38
Chapter 4 Formation and Evolution of X-ray Binaries
39
Chapter 4 Formation and Evolution of X-ray Binaries
Example: constraint on the initial system parameters of PSRs
1913+16 and 2303+46
PSR 1913+16
Porb =0.32 d, e = 0.617
PSR 2303+46
Porb =12.34 d, e = 0.658
af
m  M  M
M

e
SN:
m  M  M , ai m  M  2M
MHe=3.16 M⊙, Porb =1.45 hr
MHe=3.29 M⊙, Porb =1.92 d
1.42
M ~12 M⊙ ( M He  0.073M1 )
af
M 2cM1
1
(
)[
]
CE: ai
M2
M1  2M 2e /(  CErL ) , CE=0.5
ai/af = 161.4
a =0.81 AU, Porb =72.3 d
ai/af = 152.8
a =7.71 AU, Porb =5.84 yr
40
Chapter 4 Formation and Evolution of X-ray Binaries
(3) Formation of LMXBs
41
Chapter 4 Formation and Evolution of X-ray Binaries
Formation of LMXBs in globular clusters
(i) Tidal capture: During the close passage of a compact star a
normal star would undergo substantial tidal deformation at the
cost of a part of the relative kinetic energy of the orbit. Most
of this energy will eventually be dissipated through oscillation
and heating. If the amount of the
energy thus lost exceeds the total
positive energy of the initial
unbound orbit a bound system will
result.
42
Chapter 4 Formation and Evolution of X-ray Binaries
(ii) Exchange encounter: a compact star interacts with a binary,
and replaces one of the binary components.
43
Chapter 4 Formation and Evolution of X-ray Binaries
The rate at which stars with number density n encounter
target stars with number density nc in a cluster with dispersion
velocity v is given by
nc nR
   nc nAvdV  
dV
v
where A is the interaction cross section, R the radius of the
star.
The cross section A for an encounter with a distance of
closest approach within d contains a geometrical and a
gravitational focusing contribution
m1  m2
A  d (1 2G
)
2
v d
2
44
Chapter 4 Formation and Evolution of X-ray Binaries
An analogous equation gives the exchange encounter rate
nc nb a
e   nc nb Ab vdV  
dV
v
where nb is the number of binaries per unit volume, and a the
semi-major axis of the binary.
The ratio of tidal capture to exchange encounters is roughly
 R n
~
 e a nb
45
Chapter 4 Formation and Evolution of X-ray Binaries
46
Chapter 4 Formation and Evolution of X-ray Binaries
(4) Evolution of LMXBs
 Mechanisms driving mass transfer in LMXBs
(a) Loss of orbital angular momentum
–Gravitational radiation
–Magnetic braking
–…
(b) Nuclear evolution of the companion star
47
Chapter 4 Formation and Evolution of X-ray Binaries
(i) In low mass systems with Porb, i > ~1-2 d, the mass transfer is
driven by the internal evolution of the low-mass (sub-)giant
companion stars.
For a given metal content Z the outer radius of such stars is
uniquely determined by the core mass. For Z=0.02 and 0.16 ≤
Mc/M⊙≤ 0.45 the radii and luminosities of such stars can be
fitted by simple polynomial relations in y = ln (Mc /0.25 M⊙)
ln( R2 / R )  a0  a1 y  a2 y 2  a3 y 3
ln( L2 / L )  b0  b1 y  b2 y 2  b3 y 3
with (a0, a1, a2, a3) = (2.53, 5.10, -0.05, -1.71), and (b0, b1, b2, b3)
= (3.59, 8.11, -0.61, -2.13).
48
Chapter 4 Formation and Evolution of X-ray Binaries
There is a relationship between the giant’s radius and the mass
of its degenerate helium core, which is entirely independent of
the mass in the envelope.
Hence the final orbital period
is expected to be the
function of the mass of the
resulting white dwarf
(Rappaport et al. 1995).
49
Chapter 4 Formation and Evolution of X-ray Binaries
(ii) For systems with Porb <~10 hr, interior evolution of the
companion plays a negligible role and the evolution of system is
driven by angular momentum losses by magnetic braking and
gravitational radiation.
(iii) In the intermediate period range between ~10 hr and ~1-2
d, both angular momentum losses by magnetic braking and the
radius expansion due to the interior evolution of the subgiant
play a role.
50
Chapter 4 Formation and Evolution of X-ray Binaries
 Bifurcation period
Pylyser and Savonije (1988) found that there is a bifurcation
period Pb for the initial orbital period.
Below Pb magnetic braking wins, and
the systems evolve to shorter
periods.
Above Pb nuclear evolution wins and
systems evolve to longer periods.
For Porb,i close to Pb the orbital period
of the system may not change much
during the entire evolution.
51
Chapter 4 Formation and Evolution of X-ray Binaries
 Problems with the standard models for LMXBs
(a) the formation of LMXBs requires a contrived evolution
- extreme initial mass ratio
- ejection of a massive common envelope by a low-mass star
- survival as a bound systerm after the SN
52
Chapter 4 Formation and Evolution of X-ray Binaries
(b) Birthrate Problem (Kulkarni & Narayan 1988)
NLMXB ~ 100, τLMXB ~ 109 yr
NBMSP ~ 105, τBMSP ~ 109-1010 yr
NBMSP /τBMSP ≤ 10-100 NLMXB /τLMXB
The lifetimes of short period LMXBs may be shorter than
previously thought by a factor of 10-100.
53
Chapter 4 Formation and Evolution of X-ray Binaries
(c) Angular momentum loss by magnetic braking
Recent work by Andronov, Pinsonneault, & Sills (2001)
indicates that the standard formulae overestimate the angular
momentum loss in CVs by as much as 1-2 orders of magnitude.
Cannot explain why
the orbital distribution
of LMXBs is different
from that of CVs.
54
Chapter 4 Formation and Evolution of X-ray Binaries
(d) Pulsar Mass Problem
Most of the WD companions of BMSPs have masses of ~
0.1-0.4 M⊙.
In most cases of LMXB evolution,
the mass transfer rates are
sub-Eddington, so BMSPs should be
massive (~2.0 M⊙) neutron stars.
Pulsar mass measurements show
that M = 1.46 (±0.30) M⊙
M = 1.97(4) M⊙ for PSR 1614-2230 (Nat 467, 1681)
large mass losses during
binary evolution
the
Zhang et al. (arXiv/1010.5429)
55
Chapter 4 Formation and Evolution of X-ray Binaries
Possible solutions (?)
(a) X-ray irradiation
Irradiation-driven wind (Ruderman et al. 1989)
Irradiation-driven expansion (Podsiadlowski 1991)
(b) Different channels for the formation of MS pulsars
Accretion-induced collapse
Formation from intermediate-mass X-ray binaries
56
Chapter 4 Formation and Evolution of X-ray Binaries
(5) Evolution of intermediate-mass X-ray binaries (IMXBs)
 Prototype: Her X-1 (Porb≈1.7 d, Md≈2.35 M⊙)
Birthrate rate estimation
rHer X-1 N HerX 1 tLMXB
1 109



 5  100
rLMXB
N LMXB tHerX 1 100 10
Cygnus X-2: Product of IMXB
Evolution
Parameters:
Porb=9.84 d, L2 = 130-160 L⊙
Spectral type: A9±2 III
q = 0.34±0.04, M2 = 0.49-0.68 M⊙

57
Chapter 4 Formation and Evolution of X-ray Binaries
The phase space of Porb at onset of mass transfer versus initial
main-sequence companion mass, M2,i, of NS-MS binaries
leading to various outcomes.
58
Chapter 4 Formation and Evolution of X-ray Binaries
IMXBs are more common than LMXBs.
A significant fraction of LMXBs are actually IMXBs or their
descendants, so that IMXBs provide a new formation channel
for BMSPs.
The final Porb-MWD
relation for BMSPs
59
Chapter 4 Formation and Evolution of X-ray Binaries
 Problems:
- Invoking IMXBs does not solve the birthrate and pulsar mass
problems.
- Luminosity distribution inconsistent with observations
60
Chapter 4 Formation and Evolution of X-ray Binaries
- BMSPs with Porb > 100d generally have M2 smaller than
predicted by theory.
- There are a cluster of MSPs between 0.1d <Porb< 1d that are
inconsistent with being progeny of any class of LMXBs.
61
Chapter 4 Formation and Evolution of X-ray Binaries
-PSR J1903+0327 (D. Champion et al. 2007)
Ps=2.15 ms, Porb=95 d, e =0.44
c=1.5(2) Gyr, B =2.2(1)108 G
Mc 0.88 M⊙
Rules out the recycled pulsar theory and DNS theory.
Born as is?
Born in GC? No GC found at the position (disrupted, ejected).
Triple system?
62
Chapter 4 Formation and Evolution of X-ray Binaries
References
1. Bhattachary, D. and van den Heuvel, E. P. J. 1991, Phys. Rep.
203, 1
2.Heger, A. et al. 2002, astro-ph/0211062
3.Taam, R. E. and Sandquist, E. L. 2000, ARA&A, 38, 113
4.Tauris, T. M. and van den Heuvel, E. P. J. 2003,
astro-ph/0303456
5.Deloye, C. J. 2007, astro-ph/0710.0189
6.Ritter, H. 2008, astro-ph/0809.1800
63