Tutorial on massive close binary evolution, 27
February 2009, Ed van den Heuvel
Stellar Timescales
Nuclear Timescale: time to exhaust its hydrogen fuel:
10
-2.5
= 10 M
(M in solar units and ≥ 1)
yr
Thermal timescale: time to emit the star’s thermal energy content at its present Luminosity
2
2
According to the Virial theorem, Eth = - 0.5 Epot,grav ~ GM /R, so τ ~ GM /RL
= 3.1 x 107 M -2 yr
Dynamical Timescale = Pulsational timescale = time to restore Hydrostatic Equilibrium
= R/c sound
0.5
= 50 min (ρSun /ρ)
“CONSERVATIVE” EVOLUTION:
Simplest Case: total mass and total orbital angular momentum conserved
(in most cases rotational angular momentum of components much smaller than
orbital angular momentum (but not always!)) ; orbits are assumed circular
Combination of (7) and (8) gives:
.
If Jb = constant, then: since M2 <0: if M2 > M1 , orbit shrinks, in opposite case it widens
i = initial, f = final
Once the more massive star overflows its Roche lobe and transfers matter to its companion,
its Roche-lobe radius shrinks while its thermal equilibrium radius stays about the same; if it
has a radiative envelope the star temporally shrinks due to the mass loss, but it then expands
on a thermal timescale to restore its thermal equilibrium. As a result it continues to transfer
matter until it has become the less massive component of the system and further mass
transfer causes its Roche lobe to expand. The entire process takes ~ τ(thermal):
3
7
Mdot ~ 0.8M/τ(thermal) ~ 0.8M /(3.10 ) [Msun/yr]
for M ≥1.5Msun ,
Mdot ≥ 10
-7
Msun/yr
NON-CONSERVATIVE EVOLUTION AND THE FORMATION OF A
COMMON ENVELOPE
The above holds for primary stars with radiative envelopes. However, if the envelope
is CONVECTIVE the mass transfer will take place on a DYNAMICAL timescale.
This is due to the fact that with adiabatic convection the envelope has
γ = Cp/Cv = 5/3. Such a star is a polytrope of index 1.5, similar to a nonrelativistically degenerate White Dwarf. For such a star the Radius increases when the
mass decreases. Since the convection goes with the sound speed, this radius increase
takes place on a DYNAMICAL TIMESCALE, which means: instantaneous.
At the same time, the mass transfer from the more massive to the less massive star
makes the orbit shrink. This leads to a violently unstable mass transfer and the
formation of a Common Envelope: The secondary star now spirals down into the
common convective envelope.
The Classical Case here is (Paczynski, 1975; Ostriker 1975): the formation of the
Cataclysmic Variable (CV) binaries though Case C mass transfer and the formation
of a Common Envelope:
An AGB star with a deep convective envelope and a degenerate CO-core overflows
its Roche lobe towards a G- or K-dwarf companion, which is in a wide orbit (orbital
period: years). The dwarf companion spirals down in the envelope of the AGB star,
undergoing tremendous drag, causing its orbit to very rapidly shrink.
The loss in orbital gravitational potential energy in the process must be at least as
large as the gravitational binding energy of the envelope of the AGB star, in order to
have the Common Envelope expelled. This condition determines whether or not there
remains a binary (in a very tight orbit) consisting of the degenerate CO-core plus the
G- or K- dwarf companion.
This binding energy condition leads to a relation between the final and initial orbital
radii of the system, for example the one derived by Tutukov and Yungelson (1978) or
by Webbink (1984): See next page.
If there is insufficient orbital energy available to remove he envelope, the two stars
will merge. Whether or not this will occur depends on the orbit and on the
characteristic parameters of the two stars.
Tutukov-Yungelson formalism for
Common-Envelope Evolution
sum of initial
masses
lost envelope
mass
Efficiency-parameter of
CE-evolution
final masses of
of donor and accretor
Final orbital
radius
Initial orbital
radius
In general: initial and final mass of te donor( = inspiralling star) is the same;
Initial accretor mass = mass of the giant;
Final accretor mass = mass of degenerate core of the giant.
Some examples of the evolution of massive close
binaries:
On the next page the evolution of a binary with initial components of 12 and 6 solar
masses.
After the first CONSERVATIVE mass transfer there remains a binary consisting of a
15 Msun main sequence star plus a 3Msun helium star (the core of the 12 Msun star).
The orbital period has increased, according to equation (9).
The formula for the increase for conservative evolution is:
P/Pi = [(M1i . M2i)/ (M1f . M2f)]² , where the symbol i and f indicate the initial and
final situation of the system. Inserting the above initial and final component masses
one finds that the orbital period has increased by a factor 2.56 sue to the mass transfer.
Starting with a period of 10 days one has 25.6 days after the mass transfer.
When the 3Msun He star has exploded as a SN, there remains a relatively wide binary
consisting of a 15 Msun B-star plus a neutron star. Such a system resembles the Bemission X-ray binaries.
When the 15 Msun B-star evolves into a red giant and overflows its Roche lobe, the
neutron star will engulfed by the giant’s envelope and spiral down into the envelope.
The final result will be a very close binary consisting of the Helium core of the giant
plus the neutron star, resembling the 4.8-hour period X-ray peculiar binary
Cygnus X-3 (van den Heuvel and DeLoore, 1973). When the helium star in this
systems explodes as a supernova, one expects that either an eccentric-orbit double
neutron star, or two runaway neutron stars will result.
The B-emission X-ray binaries form the largest class of High-Mass X-ray Binaries in
the Galaxy and the Magellanic Clouds, with almost 100 known members. Particularly
the SMC has several tens of them.
That they outnumber the “Standard High Mass X-ray Binaries” such as Cen X-3, Vela
X-1 and Cyg X-1, which have O- and B-supergiant companions with initial masses >
20 solar masses, is mainly a question of the IMF.
Be-star
~ 15M sun
He star
~ 3M sun
“Standard” scenario for forming
a double neutron star, starting
from a (B-emission) X-ray binary
(e.g. Flannery and vdHeuvel, 1975
Srinivasan and vdHeuvel 1982);
NS
First-born NS is “recycled”:
it underwent much accretion,
which caused its magnetic field
to have weakened, and was
spun-up by accretion from
Be-star and second He-star.
2nd He-star
2nd NS
(Drawing from Dewi, Podsiadlowski
and Pols, 2005)
[e.g. Cyg X-3, P= 4.8h]
Double neutron star
A second example is the formation of the “Standard” High-Mass X-ray
Binaries.
Starting with a typical O-type close binary with components of 25 and 15 solar
masses, and an orbital period of 5 days, one is left after the first mass transfer
phase with a binary with components of 8 Msun (Helium star= Wolf-Rayet star) plus
32 Msun, and an orbital period of about 8.5 days.
The system then has an age of about 6 million years.
The 8 solar-mass helium star explodes as a supernova about 0.5 million years later.
Assuming that it lost in the meantime 2 solar masses due to its WR-wind, one
calculates that the orbital period after the supernova will be of order 10 days, the orbit
having become eccentric due to the about 4.5 Msun SN mass loss(also if the NS does
not receive a kick), and the system will have received a runaway velocity (“slingshot
effect”) of order 40 km/s. Such runaway velocities are indeed observed for the
standard HMXBs.
The 32solar mass O-type companion will remain another 4 or 5 million years on the
main sequence and when it moves to the supergiant stage the system becomes an Xray binary for probably not more than a few tens of thousands of years up to perhaps
hundred-thousand years.
This whole scenario works also for HMXBs with a black hole, such as Cygnus X-1
(van den Heuvel 1974, Proc. Solvay Conference “Astrophysics and Gravitation”
1973, Brussels).
This system consists of a O9.7Iab supergiant with probably a mass of ~ 25 Msun and
a black hole of about 10-12 Msun.
The progenitor of this black hole must have been a helium star of > 20 Msun, which
requires an initial mass of larger than ~ 50 Msun.
The enshrouded (obscured) flaring High-Mass X-ray Binaries discovered with the
INTEGRAL satellite: a new category
The hard-X-ray IBIS instrument of ESA’s Gamma-ray observatory INTEGRAL
discovered a new type of HMXBs with X-ray emission only at energies above 10 to
20 keV.
Apparently, there is so much gas in these systems that X-rays of lower energy are
completely absorbed. Since earlier X-ray observatories worked at E < 10-20 keV,
these systems were never seen before.
These X-ray binaries can give large X-ray flares, up to the Eddington limit, lasting for
a few hours. The X-ray sources are X-ray pulsars with very long pulse periods, often
hundreds of seconds.
The companions in all identified 8 cases are O-type supergiants. These are the
systems IGR J08408-4503, IGR J11215-5952 (Pspin= 186.8 sec), IGR J16479-4514),
XTE J1739-302, IGR J17544-2619, SAX J1818.6-1703, AX J1841.0-0536 and IGR
J18483-0311, Pspin ~ 21 sec). There are 15 more of these flaring sources, but not yet
with an identified optical star (Sidoli et al. Astro-ph 0810.5446v1 dd 30 Oct.2008).
The orbital period has so far been well established for one system: IGR J11215-5952
and turns out to be very long: 165 days (Romano, Sidoli et al. 2009, Astro-Ph
0902.1985v1 dd 11 Feb. 2009). Its neutron star has a spin period of 186.78 sec.
Two more flaring systems have likely period determinations:
IGR J16479-4514, with Porb ~ 30 days, e=0.4; and: XTE J1739-302: Porb ~ 70 days,
e=0.4 (Ducci, Sidoli et al. Astro-ph 0810.5463v1, 30 Oct.2008).
It thus appears that these enshrouded systems are all long-period binaries. The
discovery of these systems solved the problem of the “missing standard HMXBs”
with orbital periods longer than about 10 days (until the discovery of the IGR sources
only one such system was known: 4U1223-62 with P=41.5 days, Pulse period ~ 700
sec). On the basis of the orbital periods of Wolf-Rayet binaries one had expected such
systems to exist in nature.
A third example: the formation of the BlackHole Low-Mass X-ray Binaries
These systems consist of a Roche-lobe filling M-, K-, G-, F-, A- or B-dwarf (= mainsequence star) which orbits a black hole with a mass between 4 and 18 solar masses.
A prime example is the first-discovered system of this type: A0620-00 (also called
“X-ray NOVA Mon, 1917, 1975). This an X-ray “transient” and in 1985 McClintock
and Remillard (1986, Ap.J. 308,110) discovered that during its “off” phase there is at
its position a K-dwarf that orbits an unseen object in 8 hours with a very large orbital
velocity: 457 ± 8 km/s . That velocity indicates a lower mass limit for the unseen
companion of 3.20 Msun, indicating that it can only be a black hole.
Since then some 20 systems of this type have been discovered. [A list of a number of
these systems and their characteristics is given in the document “More on Long GRBs
and Starburst galaxies I, 26 february 2009”, which was a handout for my yesterday
talk].
Basically, the formation scenario of these objects is similar to that of the CVs, scaled
up to higher masses (e.g. de Kool, van den Heuvel and Pylyser, 1987,A&A 183,47 ):
A case C model in which the envelope of a massive red supergiant (> 20 Msun) with a
very evolved core (containing already a considerable fraction of Carbon and Oxygen,
etc.) engulfs a distant K- or G- (or other) dwarf companion. That companion spirals
down into the supergiant’s envelope, driving off this envelope, such that a close
binary remains consisting of the dense evolved core of the Supergiant, plus the
unaltered dwarf companion.
After the collapse of the core to a black hole, a system remains resembling the BHLMXBs.
An alternative model, suggested by Podsiadlowski, is that these systems are later
evolutionary products of BH-High-Mass X-ray Binaries such as Cygnus X-1 (orbital
period 5.6 days). When in such a system the supergiant overflows its Roche lobe, a
Common Envelope will form and in these short-period binaries the compact star will
spiral down into the core of the companion, causing a “merger”. Due to the high
angular momentum (the orbital angular momentum is now deposited in one star),
it is likely that the resulting object will be a black hole surrounded by a large disk,
resembling a proto-planetary disk.
In this disk, small stars can condense out, and after the disappearance of the disk, a
binary remains that resembles the BH-LMXBs.