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Physics seminar, Department of Physics, IIT Kharagpur

October 3, 2023

Observable properties of massive Algols binaries

Koushik Sen

Institute of Astronomy, Nicolaus Copernicus University, Torun

Collaborators: N. Langer, D. Pauli, G. Gräfener, A. Schootemeijer, H. Sana, T. Shenar, L. Mahy, C. Wang, S. de Mink

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Massive stars

Mass greater than 8 M

2

O type star

Early B type star

More massive stars are larger in radius
More massive stars have shorter lifetimes

Mass is fundamental in stellar evolution

Radiative envelope

Convective core

Now, what are massive stars? Massive stars are objects that are typically heavier than eight times the mass of our Sun. Their structure usually consists of a convective core where nuclear fuel is burning, and a radiative envelope where the energy generated inside the core is radiatively transported to the surface. This figure shows the radius evolution of a fifteen solar mass star as a function of its age. The radius gradually increases while the star is burning hydrogen at its core, then there is this huge increase in radius in a very short amount of time and spends the rest of its life there burning helium, carbon, etc. One thing to note here is that the star spends most of its life burning hydrogen, which we call the main sequence lifetime of the star. What I now show is the radius evolution of two more stars with two different masses, one greater and one smaller than fifteen solar masses. Two things to note are that more massive stars are larger in radius but have smaller lifetimes. For example, this fifteen solar mass star has a main sequence lifetime greater than the entire life of the twenty solar mass star. One more thing that you can derive naturally from this discussion is that mass is a very important parameter in stellar evolution. These lines of different masses are well-behaved and typically do not intersect with each other. Finally, stars above sixteen solar masses are called O-type stars and stars between ten to sixteen solar masses are early B-type stars.

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Multiplicity of massive stars

Massive stars are mostly born in binaries (Sana+2012,2013)

Almost all O stars will interact during their lifetime (Moe+2017)

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Fraction of interacting binaries

Sana+2012

Moe+2017

Mass M₁ (M )

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A decade of gravitational wave observations

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CC: Kim Martineau

~ 100 detections

O1

O2

O3

O4

O5

No of detections

~ 10 detections

~ 10,000-100,000 detections

LIGO runs

This schematic diagram shows one of the binary evolution channels towards the formation of compact object binaries. With ~90 compact object mergers detected, a lot of people are interested in this final stage. But in all this excitement, I think we have overlooked a lot of uncertainties in the stages leading up to the compact object binary stage. To get accurate predictions of the double compact binary stage, we need to make sure that our models can explain the observed binaries in earlier stages, where both the observed sample size is larger to compare to and, we have more precise measurements.

But do we need to understand all this? What do real stars do? It has been empirically found that more than 50% of massive stars will interact with each other during their lifetime. Most importantly, for short-period binaries, there is a nuclear timescale mass transfer phase where we can observe binaries in this semi-detached configuration, commonly known as Algol binaries. It has also been shown that rapid binary evolution codes, that make severe approximations to the structure of the star, cannot accurately model binary evolution. From these binaries, we can try to get a handle of the mass transfer efficiency in binaries.

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The first results…

Formation of gravitational wave progenitors from massive binaries

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Tiwari+2023

LIGO/VIRGO

Mass distribution of LIGO black holes

Spin distribution of LIGO black holes

This schematic diagram shows one of the binary evolution channels towards the formation of compact object binaries. With ~90 compact object mergers detected, a lot of people are interested in this final stage. But in all this excitement, I think we have overlooked a lot of uncertainties in the stages leading up to the compact object binary stage. To get accurate predictions of the double compact binary stage, we need to make sure that our models can explain the observed binaries in earlier stages, where both the observed sample size is larger to compare to and, we have more precise measurements.

But do we need to understand all this? What do real stars do? It has been empirically found that more than 50% of massive stars will interact with each other during their lifetime. Most importantly, for short-period binaries, there is a nuclear timescale mass transfer phase where we can observe binaries in this semi-detached configuration, commonly known as Algol binaries. It has also been shown that rapid binary evolution codes, that make severe approximations to the structure of the star, cannot accurately model binary evolution. From these binaries, we can try to get a handle of the mass transfer efficiency in binaries.

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6

~ 1000 binaries

~ 1 merger

Your favorite stellar evolution code + physics assumptions

Reality

More than 100 papers predicting merger rates!! (Mandel+2022)

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From stars to black holes

“mass transfer efficiency” (Langer+2003, de Mink+2007,Sen+2022).
Wolf-Rayet wind mass loss (Grafener+2011; Sanders+2022)
Supernova “kicks” (Mirabel and Rodrigues 2003, Dhawan+2007; Repetto+2012, Belczynski+2016).
Black hole (BH) formation (Sukhbold+2016, Ertl+2016, Woosley+2020).
Wind-fed High Mass X-ray Binary (HMXB) formation (Shapiro and Lightman 1976, Iben and Tutukov 1996, Vanbeveren+2020, Sen+2021).

7

Kruckow+2018

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Binary evolution modelling

1D stellar evolution code
Individual binary components are evolved simultaneously
Details of mass transfer is calculated at each timestep
Essential for studying interacting binaries

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Detailed binary evolution with MESA (Paxton+2011,13,15,18,19)

Finally, a few words on how we can model binaries. There are two approaches that we take here. One is called rapid binary evolution and the other is detailed binary evolution. As the name suggests, in rapid binary evolution, individual stars are taken as point objects or approximated by single star models of hydrogen and helium burning stars. Then, we use fitting formulas to estimate the outcome of a mass transfer phase once one of the stars is expected to fill its Roche lobe. Due to these approximations, the calculations can be done fast and these are typically useful to study the properties of binaries before they have interacted, or after all the interaction phases are over. Based on the approximations used, there are a number of codes available in the market, such as COMPASS, COMBINE, COSMIC. Etc. On the other hand, detailed binary evolution codes evolve both stars simultaneously and the details of the mass transfer phase are calculated to great precision at every timestep of the evolution. So, this approach is slow, but if you want to study interacting binaries, which in my opinion are also the most interesting, you need to use this method. One of the most widely-used, open-source stellar evolution codes used by the community is MESA, and this is what I also used in my thesis.

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PART I: MASSIVE ALGOL BINARIES

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Arabic: رأس الغول, Ra’s Al-ghūl, Head of the Demon

BATMAN

ECLIPSING BINARIES

Porceddu+2008, Jetsu+2013,2015

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Algol binaries and the Algol paradox

Algol or 𝜷-Persei, is a star system that shows periodic variations in brightness

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The Algol paradox: Less massive star is more evolved than its companion
Resolution: Mass transfer in close binaries (Pustylnik+1998)

Credit: SpaceEngine

M₁ = 0.7 M , R₁ = 3.48 R

M₂ = 3.2 M , R₂ = 2.73 R P_orb = 2.87 days

Montanari1671, Goodricke1783, Pickering1881

Algol, or Beta Persei is the prototype star system for these type of binaries. Algol is also one of the first stars in ancient astronomy that was recorded to show periodic variations in its brightness. Due to this change in brightness, ancient astronomers associated it with witchcraft and demons. Indeed, the term Algol is actually derived from the Arabic word Ra’s al-ghul, which means the head of a ghoul. Modern day astronomy has of course shown that the change in brightness has nothing to do with demons, but is because the viewing angle from Earth to the system is such that the stars get eclipsed during their motion in the orbit. Further studies showed that the currently less massive star of the system is actually larger in radius. If you assume that the two stars have never interacted with each other, this brings us to the Algol paradox that the less massive star is more evolved than its companion. This challenged the-then theories of stellar evolution and if you remember the discussion two slides ago, the resolution lies in the fact that these are are currently interacting with each other. Indeed, if you calculate the Roche lobe radius of the less massive star for this orbital period, you will see that the less massive star is actually filling its Roche lobe. But since the mass transfer rates in this configuration are very low, people failed to recognise any direct observational evidences of this interaction then.

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Close binary evolution

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M₁ = 20 M M₂ = 12 M

P_orb = 3 d

Images credit: Philip D. Hall

M₁ = 20 M M₂ = 12 M

M₁ = 12 M M₂ = 12 M

M₁ = 8 M M₂ = 12 M

a

“Algol binaries”

This brings us to close binary evolution. Consider two stars with masses twenty and twelve solar masses, orbiting each other every 3 days. Now, to understand when these stars will interact with each other, we need to draw some equipotential lines around them. In 3 dimensions, these will be equipotential surfaces. Of these lines, one of them, in the shape of figure 8, is very interesting. This is called the Roche potential surface and the two lobes covering the two stars are called their individual Roche lobes. If a mass element is to the left of this intersection point, it will be attracted more strongly by the left star, and vice versa. During the evolution, the more massive star will increase in radius faster and at some point, just overfill its Roche lobe. At this point, the surface of the more massive star is gravitationally attracted more strongly by the other star than itself. And this starts the transfer of mass from the more massive star to the less massive star. For now, let us assume that all the mass that is transferred from the mass donating or donor star is able to end up on the mass accreting or accretor star. From simple mass and angular momentum conservation, it can be shown that the orbital period, and in turn the orbital separation through Kepler’s third law, shrinks. Now, the tendency of massive stars to increase in radius and the decreasing orbital separation will lead to increased mass transfer rates, that is, a lot of mass will be lost by the donor very quickly. This will continue until the mass ratio, here defined by the ratio of the mass of the accretor to the mass of the donor, reaches unity. After the mass ratio inverts, any more mass transfer will lead to an increase in orbital separation and hence the mass transfer rate will start to decrease from its peak value. However, as long as the radius expansion of the donor is faster than the increase in orbital separation due to mass transfer, the fast mass transfer continues. Only after most of the radiative envelope of the donor is stripped, do the two rates of expansion become close to each other. Then, you have this configuration here where the currently less massive star is filling its Roche lobe and the mass transfer rate is small, of the order of the mass divided by the lifetime of the star.

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Why study Algols?

On the main sequence -> least assumptions & uncertainties.

Nuclear timescale mass transfer -> expect many observable systems.

Short periods -> eclipsing -> easily detectable.

Stripped donor -> surface abundances -> internal mixing

Succeeds the fast mass transfer phase -> mass transfer efficiency

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M₁ = 8 M M₂ = 12 M

mergers

high-mass X-ray binaries

double compact object binaries

Algols

Pols+1994,Vanbeveren+1998,Nelson+2001,deMink+2007

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Algol binaries: Method

10,000 detailed binary evolution models (MESA, Marchant2017)

M_d,i = 10 – 40 M_Sun, q_i (M_a,i/M_d,i) = 0.275 – 0.975, P_i >= 1.4 d

Population synthesis of massive Algol binaries in the LMC

Constant star formation and empirical binary distribution functions

Compare with observed massive Algols in the LMC and Milky Way

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Mass transfer model

Ꞷ_crit = (GM/R³)^0.5

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M, R, I, Ꞷ_i

ΔM?

ΔJ = (GMR)^0.5 ΔM

ΔM/M = 𝒇 (r_g², Ꞷ_i/Ꞷ_crit)

IꞶ

ΔM/M = 0.00-0.10 (Packet+1981)

I = r_g² MR²

In binaries, there are tidal forces!

Finally, how do we decide in the models whether the mass that is lost from the donor gets accreted by the companion, that is, the mass transfer efficiency? Most works in the literature usually approximate this mass transfer efficiency by a constant. In these models, however, we investigate the effect of a more physically motivated mass transfer scheme. Let’s consider a star of mass M, radius R, a moment of inertia I, rotating at an angular velocity omega. Here, R_g is the radius of gyration of the star. Now, how much mass, delta M, having an angular momentum delta J can be accreted by this star before it reaches critical rotation? Critical rotation velocity is the angular velocity at which the centrifugal force at the equator of the star is equal to the gravitational force. Once the star reaches critical rotation, we assume that the star cannot accrete more mass.If you equate the initial and final angular momentum, you can write the delta M over M as a function of the radius of gyration and the ratio of initial angular velocity to the critical velocity. For typical values of R_g of stars, it has been shown that the amount of mass needed to be accreted is only a few percent of the total mass of the star. In conclusion, it is very easy to spin up a star and mass transfer is inefficient. Fortunately, there is another force, namely the tidal force that can halt the spin-up of the accreting star in close binaries. The strength of the tidal interaction is quantified by this tidal synchronisation term here. For small orbital separation, the tau_sync is very small, then the tidal force is very strong, such that stars that are close to each other have their spin angular velocity locked to the orbital angular velocity of the binary. In such a case, you can have an efficient mass transfer in our models.

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Algol binaries: Results

Models predict high mass transfer efficiency at P_orb < 5 days and vice versa

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Sen+2022A&A...659A..98S

Evidence of efficient mass transfer in short orbital period massive Algols

Observations: Ostrov+2000,2001, Malkov2020, Mahy+2020a,b

This 2D grayscale histogram shows the orbital period-mass ratio distribution of the models during the Algol phase. The models predict a concentration of massive Algol binaries at an orbital period of around 2 days and a mass ratio of around 2. There is also a dependence of the maximum mass ratio on the orbital period. This actually captures the spin-dependent mass transfer algorithm in the models, where the mass ratio can go as high as 3 at very short periods where the mass transfer is efficient. At higher orbital periods, the maximum mass ratio reached is much lower. The different coloured stars show the observed massive Algols in the LMC and the red circles show the ones in the Milky Way. At first glance, there is not enough observed systems to compare to the three orders of magnitude covered in our predictions, and there is a lot more scatter in the observed data than is predicted by the models. There should be at least 10 times more data to derive strong conclusions that will stand the test of time. Nevertheless, let us analyse the plot by dividing it into subareas. At the shortest orbital periods, it is likely that the observed Algols also underwent an efficient mass transfer as is predicted by our models. In the next bin, the Algols in general have a lower mass ratio than the previous bin, indicating that some of them may have undergone less efficient mass transfer. In the next two bins, there are very few data points and here even the two Algols seem to have undergone more efficient mass transfer than predicted by the models. So, there is some evidence of an orbital period-dependent mass transfer efficiency from the observed Algols, though we need much more data, especially at the higher periods to see if the dependence is monotonic or diverges away from the model predictions.

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Algol binaries: Takeaways from Part I

Mass transfer is efficient at short orbital periods.
Spin-dependent accretion can produce an orbital period dependent mass transfer efficiency.
Around 90 O-type and early B-type Algol binaries in the LMC.

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~ 50 more potential massive Algol binaries are being observed in the Milky Way using the Mercator telescope

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PART II: REVERSE ALGOL BINARIES

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Semi-detached binaries in VFTS-TMBM*

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Name of system	Donor mass (Msun)	Accretor mass (Msun)
VFTS 652	6.5	18.1
VFTS 061	8.7	16.3
VFTS 450	27.8	29.0
VFTS 176	28.3	17.5
VFTS 094	30.5	28.2

*Evans+2011, Almeida+2017, Mahy+2020a,b

Algol binaries

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If I can’t see them, I will be fine :P

Model predictions vs massive Algols

Sen+2022

Observations: Ostrov+2000,2001, Malkov2020, Mahy+2020a,b

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Happy me after publishing a major portion of my thesis

VFTS 176

VFTS 094

Name of system	Donor mass (Msun)	Accretor mass (Msun)
VFTS 176	28.3	17.5
VFTS 094	30.5	28.2

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M_donor,i = 44.6 M_sun

q_i = M_accretor,i/M_donor,i = 0.5

P_orb,i ∼ 7.9 days

Ratio of core mass to envelope mass in massive stars increases with stellar mass

Donors may remain the more massive component!

Ratio of core mass to envelope mass in massive stars increases with stellar mass

Donors may remain the more massive component!

M_donor,i = 44.6 M_sun

q_i = M_accretor,i/M_donor,i = 0.5

P_orb,i ∼ 7.9 days

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Algol configuration at higher masses

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Sen et al. (2023)

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Reverse Algol evolution

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donor

accretor

O star-like spectrum

can show emission lines!

Sen et al. (2023)

Hydrogen-rich Wolf-Rayet star on the main sequence

Well, if you want to explain stars in the main sequence, the obvious thing to do is study binaries that undergo mass transfer on the main sequence. This HR diagram shows the evolution of a massive binary with an initial donor mass of 44.6 solar masses, an initial accretor mass of 22 solar masses and an initial orbital period of 8 days. Now, the tracks are coloured red for the donor and blue for the accretor as long as the donor undergoes core hydrogen burning. The remaining evolution is shown in grey for both. During this pre-interaction phase, the donor behaves like an ordinary O star. The dots are placed every 50000 years. Mass transfer on the main sequence occurs first at the thermal timescale, where the donor is stripped up to the outer edge of the H-He gradient region such that the surface hydrogen mass fraction remains close to its initial value. Simultaneously, the donor’s luminosity remains similar but has lost much mass. So, its L/M ratio increases. This, in turn, also increases its wind mass loss rate, which can lead to an optically thick wind. Then, the donor spends its remaining main sequence lifetime as a stripped star with a relatively high surface hydrogen mass fraction. It can show a WR-like spectrum at luminosities where ordinary single stars will not.

As a side note, this kind of evolution falls into the broader category of what I now call reverse Algol binaries, where the donor stays more massive and more luminous than the accretor, unlike their lower mass Algol counterparts.

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Ordinary O stars vs Wolf-Rayet stars

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O star spectrum

WR star spectrum

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Wolf-Rayet stars

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Hainich+2014

MS single stars

lower luminosity, high surface H

X_H >= 0.4

The right panel shows the position of WR stars in the Large Magellanic Cloud on the Hertzsprung-Russell diagram. More than half of them are hotter than the hydrogen zero-age main sequence and can be largely explained as products of post-main sequence mass loss in single stars or binary interaction. It is important to note here that post-main sequence wind mass loss or binary mass transfer cannot reasonably explain the WR stars that are on the main sequence band here because those models shoot back across the main sequence to higher temperatures and low surface hydrogen levels at the thermal timescale.

The next thing I want you to notice is the surface hydrogen mass fraction of these ms-WR stars. The ones on the main sequence band seem to have higher surface hydrogen. Indeed, when I plot the ones with surface hydrogen greater than or equal to 0.4, I find most of them lie on the main sequence band. While the most luminous ones can still be explained by ordinary single stars where the wind mass loss rates are high enough for them to form an optically thick wind, this handful of low luminosity WR stars exists, and some even show a surface hydrogen mass fraction of 0.7. I hope I have convinced you that the low luminosity WR stars cooler than the ZAMS may be different beasts and hence are formed via alternative channels.

This brings me to the two questions I want to answer: 1) How are these H-rich WR binaries formed on the main sequence? 2) Can these apparently single H-rich ones be undetected binaries, and if they can be, how can we detect their companion?

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Wind optical depth parameter

Differential form of the wind optical depth (de Loore+1982)

d𝜏 = - 𝜅𝜌(r) dr

dM/dt = - 4𝜋r²𝜌(r) 𝜐(r)

𝜐(r) = 𝜐₀+(𝜐_∞- 𝜐₀)(1-R_*/r)^𝛽

For O stars, 𝛽 = 0.8-1 (Vink+2000)

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Langer+1989, Aguilera-Dena+2021

Now, to quantitatively determine if our stripped donors can show an emission spectrum, the differential equation that defines optical depth can be integrated for a beta velocity law and the continuity equation. For beta equals 1, it takes the form of this equation where kappa is the electron scattering opacity dependent on the surface hydrogen mass fraction, Mdot is the wind mass loss rate, Rstar is the radius of the star, Vinfinity is the terminal velocity of the wind and v0 is the wind launching velocity at the base of the wind. Now, this quantity is what I can an optical depth parameter. While this is definitely not the true optical depth for which I will need to do stellar atmosphere modelling, it has the advantage that all the quantities on the right hand side of this equation are available both for the observed stars and our models. So, we can use this parameter to compare our models with observations “consistently”.

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Wind optical depth parameter

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Sen et al. (2023)

t [Myr]

pre-interaction

fast Case A

stripped donor

end of H burning

4.2

4.4

4.6

4.8

Here, we put it all together. The thick black line shows the evolution of the optical depth parameter of the donor model as a function of age. This vertical line denotes the end of the hydrogen burning of the donor. The pre-interaction phase is here, where the optical depth is around 0.02-0.03, and the donor behaves as an ordinary O star. But, during the fast Case A phase, the donor loses a lot of mass, L/M increases, the wind mass loss rate increases, and there is almost an order of magnitude jump in the optical depth parameter. The value keeps increasing during the stripped donor phase. The blue horizontal lines here show the estimated value of the optical depth parameter for the observed H-rich Of and Of/WN type stars. The red horizontal lines show the same for the H-rich WN stars. We clearly see that the optical depth parameter of the donor is in the correct range during the stripped donor phase. Hence, we expect our donor models to show a WR-like spectrum.

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Our smoking gun - BAT99 113

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Sen et al. (2023)

observed

model

single

Köhler+2015

M_WR= 52 Msun, M_OB= 20 Msun

P_orb= 4.5 d, X_H = 0.7

Langer+2014

Next, I show the best-fitting binary model to the observed H-rich WN+O binary BAT99 113. The dynamical mass estimate of the observed WR star is about 52 solar masses, and its companion is about 20 solar masses. A red triangle marks the WR star, while a blue diamond marks the companion. On the HRD, we see that the donor goes through a nuclear timescale evolution near the observed position of the WR star. To rule out the pre-interaction phase of the model, we also compare the L/M ratio of the model to the observations, which is defined this way. The curly L/Lsun ratio of the observed WR star is 4.41, while that of our model during the stripped donor phase is 4.37 and much lower during the pre-interaction phase. Moreover, we found that the single-star models of Kohler have a maximum curly L/Lsun of 4.2. For reference, the curly L/Lsun ratio near the Eddington limit is 4.6, so these differences in 0.2 dexes are significant.

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Reverse Algols: Takeaways from Part II

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High mass Algols defy conventional wisdom of binary evolution.

Stripped donors on the main sequence can resemble Wolf-Rayet stars.

Defining observables: high L/M, enhanced winds, emission lines

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Era of bin. pop. syn. studies

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Kruckow+2018

initial mass distribution

(Salpeter+1958)

binary distribution functions

(Sana+2012, Moe+2017)

mass transfer efficiency

O star winds

WR mass loss

(Sander+2022)

common envelope efficiency

(Ivanova+2012)

LBV winds

(Grassitelli+2018)

supernova kicks

(Mueller+2022)

NS/BH formation

(Woosley+2019)

BH spins

(Fishbach+2021)

WR+BH binaries

(Xu,..,Sen+ in prep)

HMXBs

(Kotko+2023)

Accretion disks around BHs (Sen+2021)

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Other collaborations

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Contact binaries

High-mass X-ray binaries

GW population synthesis

A. Menon N. Langer I. El Mellah C. Belczynski J. P. Lasota C. Done X.T. Xu F. Broekgaarden

IAC, Tenerife Uni. of Bonn ESO, Chile CAMK, Warsaw Durham Univ. Uni. Bonn JHU, USA

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Eddington factor

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Sen et al. (2023)

Bestenlehner+2014

pre-interaction

fast Case A

stripped donor

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PART III: WIND-FED BLACK HOLE HIGH-MASS X-RAY BINARIES (HMXBs)

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BH high-mass X-ray binaries: Context

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Average lifetime of stripped+O star phase (0.4 Myrs) < BH+O star phase.
BH-BH merger rates (Mennekens+2014; Belczynski+2016)

Stripped star

collapse

Kruckow+2018

Detailed binary population synthesis predicts that 3 out of every 100 hydrogen-burning stars host a stellar-mass black hole. On top of that, if you recollect that more massive stars have shorter lifetimes, the lifetime of the stripped star stage is much smaller than the BH+O star phase. So, if you assume constant star formation in a volume-limited sample, you should find more BH+O star binaries than binaries that have a stripped star. Instead, there are about 80 stripped star binaries observed in the Milky Way, while there is only one system that hosts a black hole: our very famous Cygnus X-1. To resolve this problem, it has been recently suggested that when black holes form, they receive recoil velocities high enough that the binary itself gets disrupted. This however means that the predicted BH-BH merger rates in gravitational wave events will get very low. But there is this step from here to here that needs to be addressed accurately.

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BH high-mass X-ray binaries: Method

Observed sample of 18 stripped+O star binaries (Vanbeveren+2020).
Assume the stripped stars collapse to form BHs without recoil.
Follow the O star evolution using stellar models (Ekstrom+2012).
When will a BH+O binary be observable as an wind-fed High Mass X-Ray binary?

A. Formation of an accretion disk

B. X-Ray flux above detection threshold

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Formation of an accretion disk

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R_O

R_acc

Matter is accreted from the stellar wind (v_wind) of the O star

Accretion radius of the compact object (Davidson+1973)

Specific angular momentum (j) of the accreted matter (Shapiro+1976)

figure not to scale

a

So, when can a wind-fed accretion disk form? This diagram shows an O star with radius Rzero separated from the BH by a distance a. The matter accreted is from the stellar wind of the O star, where the wind velocity is given by this equation. Mzero is the mass of the O star. The BH can accrete matter from within an accretion radius definition as the region where the kinetic energy of the wind matter is smaller than the absolute value of its potential energy in the gravitational field of the BH. If the BH was stationary, then the denominator would be equal to the wind velocity, but since it is in an orbit, we have to take the vector sum of the wind velocity and orbital velocity of the BH. Now, the wind velocity across the accretion radius is not exactly the same as it depends on the exact distance. So, there will be a net influx of angular momentum through the accretion radius which can be calculated by a neat integration over the entire surface. Then, the amount of accreted angular momentum by the BH is given by this formula, where eta is an efficiency factor and omega orb is the orbital angular velocity of the binary. From detailed hydrodynamical simulations, eta is about one-third for BHs accreting matter from a stellar wind.

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Formation of an accretion disk

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This accreted matter can go in a Keplerian orbit of radius R_disk with �velocity v_disk= j/R_disk

Equating centrifugal force to gravitational force, R_disk= j² / GM_BH

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Formation of an accretion disk

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This accreted matter can go in a Keplerian orbit of radius R_disk with velocity v_disk= j/R_disk

Equating centrifugal force to gravitational force, R_disk= j² / GM_BH

An accretion disk can form when: R_disk > R_ISCO where R_ISCO = 6GM_BHƔ/c²

[ 𝛾_± = ⅙, 1, 3/2 ]

Hence, an accretion disk can form if this Rdisk is greater than the radius of the innermost stable circular orbit Risco. Risco in turn is given by this form here, where the gamma factor takes into account the spin of the BH w.r.t. the accretion disk. For a non-rotating Schwarzschild BH, gamma is 1. For a maximally rotating black hole whose angular momentum is aligned to the accretion disk, it is one-sixth. For a maximally rotating black hole whose angular momentum is in opposite direction to that of the accretion disk, it is 3/2. From stellar evolution calculations, the first formed BH is expected to be slowly rotating i.e. gamma = 1. You can immediately see that accretion disk formation is easier for the case of maximally spinning BHs with their angular momentum vector aligned to the accretion disk.

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X-Ray luminosity (L_X)

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-- mass accretion rate (Davidson & Ostriker 1973)

-- X-ray luminosity (Frank+2002, El Mellah 2017)

-- The L_x > 10³⁵ erg s^-1 to be observable (Wood+1984, Bradt+1991)

To calculate the X-ray luminosity, we first need an estimate of the mass accretion rate. For this, we note that the O star wind at a distance 'a' falls on a surface area 4pi times a squared and the surface area inside the accretion radius is pi times Racc squared. This is what the first two terms here combine to, and vrel over wind is a term that comes because, again, the BH and O star are orbiting around each other. This form of the equation is valid in binary systems where the wind speed at the binary orbital separation is larger than the orbital speed. Ignoring minor relativistic corrections, such a disk around a BH mostly radiates in X-rays, and the maximum associated luminosity is a fraction of the rest-mass energy obtainable from the mass accretion rate. For a non-spinning BH, this turns out to be one-twelfth of Mdot_acc times Csquared. As a side note, throwing matter onto a BH is the most efficient way of converting mass into energy, about one order of magnitude more efficient than hydrogen burning inside the core of massive stars. Using this luminosity estimate and accounting for the distance of each stripped+O star system from Earth, we calculate the X-ray flux at Earth and set the limit for the HMXB to be observable if the flux is greater than 10 to the power -11 erg per second per centimetre squared, which is the typical flux cut-off for non-focussing X-ray telescopes. So now, we have all the ingredients ready! When these two criteria will be satisfied during the BH+O star phase, we can observe the BH+O binary as an HMXB.

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BH high-mass X-ray binaries: Results

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Black hole spin

Sen+2021A&A...652A.138S

For Cygnus X-1

M_{O star}= 40.6 M , M_BH = 21.2 M

Orbital period = 5.9 days

Non-rotating BH: R_disk / R_ISCO = 0.20

Maximally rotating: R_disk / R_ISCO = 1.23

Now, let us look at one individual system WR31 and how our assumption on the BH spin affects the formation of an accretion disk. From this left plot, we see that Rdisk/Risco is above one for the longest time for gamma = ⅙ i.e. for a maximally rotating BH where the spin of the accretion disk is aligned to the BH. The corresponding X-ray luminosity, plotted in the lower panel, is also larger. This means that the probability of observing BH+O binaries as HMXBs increases quite significantly for the case of maximally spinning BHs. From the observational side, all three wind-fed BH HMXBs detected so far have been measured to be nearly maximally spinning. This fits perfectly with what we find about the spin of the BH. For example, if you take the case of Cygnus X-1, you will indeed find that an accretion disk does not form if the BH was not spinning while an accretion disk can form for the observed spin configuration.

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BH high-mass X-ray binaries: Takeaways

BH+O star binaries are X-ray silent for most of their lifetime.
Bias towards observing maximally rotating BHs in HMXBs.
Recoil velocity not necessary to explain number discrepancy.

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Outlook

A large spectroscopic survey of massive Algol binaries in the LMC and Milky Way will provide more constraints on binary evolution.
Large population of X-ray quiet BH+O star binaries still to be discovered (Langer+2020, Shenar+2022, Mahy+2022).

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THANK YOU

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‘BONN’YWOOD presents

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‘BONN’YWOOD presents

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