Materials: Onno Pols’ lecture notes Chapter 7,8, Tauris & van den Heuvel book chapter 4.
We have already seen in the second lecture that, in the Universe, binary systems composed of two stars orbiting each other are common, the higher the mass of the stars the more so (see for example Offner et al. 2023). Historically, this fact has been leveraged to design empirical dynamical mass measurement which rely only on Kepler’s laws to obtain masses.
However, often enough, stars in binaries are so close to each other than as they evolve, they will “touch” each other and exchange mass. In today’s lecture we will discuss in more detail how being in a binary can lead to dramatic consequences for the evolution of both stars in the system, starting with the Roche model of the potential.
- Q: We will see this in more detail in the coming in class activity, but you can already answer based on your 1Mo stellar models: what happens to the stellar radius as a typical star evolves?
- Q: Which stars evolve faster: higher or lower mass?
To describe the gas of the stars in a binary system subject to the gravity of both stars is effectively a three body problem, hard to solve even before we fold in anything else but gravity.
A useful approximation often used is the Roche model of the gravitational potential. This is based on the following key assumptions:
- e=0: the mutual orbit of M1 and M2 is circular;
- M1 and M2 are treated as point masses;
- M1 \geq M2 \gg m: the masses of the two stars are much larger than the test mass m we consider to map the potential;
These defined the restricted circular three body problem. Of these assumptions the most restrictive is perhaps the first: binaries can be eccentric, and updates to this model to account for that are actively being developed (e.g., Sepinsky 2009). The second assumption is not critical, since the density increases steeply moving inwards in the star: most of the mass is in the center. The last assumption is typically easily verified as long as one considers small parcels of gas.
With these three assumptions, one can put themselves in a frame centered in the binary center of mass and co-rotating with the binary at frequency n=2π/P. Since it is a rotating frame, it is non-inertial, but the gravitational potential of the two stars can be written as:
Where ri is the distance to Mi (considered a point-mass), so the first two terms represent the gravity of each star, respectively, and the last term represents the centrifugal force.
Taking the gradient, the gravitational force from the combined masses of the stars in the binary is F=-∇ Φ(\mathbf{r}). But because we are in non-inertial frame, there is also another term to add to the momentum equation which cannot be expressed as a potential, because it depends on the velocity of the test mass m: the Coriolis force ∝ 2n× v. Per unit volume the momentum conservation is thus
Finally, one could also relax the hypothesis 2. above and include a tidal term due to the finite size of the stars in the potential (this again works because most of the mass of the star is in it’s core, saving the Roche model, and the tides are a relatively small perturbation on top of this):
On the line connecting the two binaries, this potential has three zeros, meaning points where there is no net (gravitational+centrifugal) force. All three are saddle points and they are called Lagrangian points:
- L1 sits in between the two stars, and it is the point where the gravitational pull from each star cancel each other
- L2 is the point behind star 2 (many telescopes, including for example JWST or Gaia sit close to the L2 point of the Earth-Sun binary system)
- L3 is the point behind star 1.
N.B.: The position of the center of mass depends on the mass distribution (here the masses are assumed to be point like), while the position of the Lagrangian points depends on the dynamics (gravity and centrifugal forces)
The equipotential shapes are close to spheres for r1 and/or r2 \ll rL_{1}, then they get progressively elongated along the axis connecting the stars, and flattened by the centrifugal potential.
For short period binaries, P<10 days (though the exact value depends
on many things, especially mass M1, mass ratio q=M2/M1, metallicity,
etc.), we observe that the stars co-rotate with the orbit, that is
there is complete tidal synchronization. In that case, in the
co-rotating frame, the gas of both stars is still,
that is equipotential surfaces are also isobaric surfaces (∇\Phi = 0 ⇔ ∇ P=0), and since ρ ∼ dP/dΦ has to remain finite, it is also constant on these surfaces!
Therefore, for a tidally synchronized binary the stellar gas will be shaped like equipotential surfaces in the Roche potential, producing non-axisymmetric, tear-drop-shaped stars.
For larger periods, tidal synchronization may not occur (tides are a steep function of R/a with R stellar radius and a orbital separation), and in that case the rotational bulge may lag behind the orbit, adding an extra deformation.
On top of the necessary hypothesis (e=0, M1 ≥ M2 \gg m, point masses and in co-rotation) to derive the Roche approximation, it is worth noting that for r\gg rL_{2} the Roche model becomes less and less physically meaningful. By construction, it is based on the assumption of co-rotation with the binary, but a parcel of gas significantly outside L2 will not maintain this co-rotation, an it is more likely to only “see” the inner binary as a point mass with total mass M1+M2.
The Roche potential is clearly not spherically symmetric: how can we use this in stellar evolution simulations?! In reality, thanks to the fact that in depth, the potential is very closed to spherically symmetric, and most of the deformation is in a large by radius by extremely small in mass region of the star (nevertheless important for atmospheric effects and observable predictions!). So for the stellar interior calculation, the spherical symmetry is still acceptable for most of the mass domain and as long as the stars are detached (see below).
Each Roche lobe can then be interpreted as the sphere of gravitational influence of each star.
What we need to know then is the volume of each Roche lobe, and we can then define a sphere that has the same volume and compare the volume of the stellar gas to the volume of such sphere. There are multiple formulae to fit the Roche volume as a function of the binary parameters (e.g., Paczsynki 1971), but probably the most common one is Eggleton 1983’s formula, which provides a fit accurate to ∼1% and continuous across a large range of mass ratios:
where a is the orbital separation, qi = Mi/Mj, and 4π RRL,i3/3 ∼ Roche volume of star i.
As we will see in the next in class activity, stars tend to grow bigger as the evolve: even a binary system that stars as detached may come into contact as the stars evolve, triggering the onset of mass transfer between the stars in the binary system. In fact, gas reaching L1 will be equally bound to either stars, and it can easily be perturbed to fall into the other star (see also review by Marchant & Bodensteiner 2024).
Depending on the stability of the orbital response to the transfer of mass, we distinguish two cases: stable Roche lobe overflow or dynamically unstable common envelope.
If Roche lobe overflow does not hit a runaway response of the orbit (causing more and more overflow), then it is dynamically stable.
This will occur in the majority (50-70%) of massive stars, and it can have important consequences for both stars that will be modified by this interaction (see below).
Sometimes, the orbit and or the evolution of the stars respond to mass transfer increasing the amount of overflow, leading to an unstable situation: in this case the system enters in contact first, and then ultimately in a common envelope event (see reviews by Ivanova et al. 2013, Ropke & de Marco 2023 and Ivanova et al. 2020s book)
During a common envelope the gas of the envelope of both stars fills equipotentials beyond L2, cannot maintain co-rotation, and thus start exerting a friction on the orbit of the two cores (or core and star) inside this shared envelope.
This results in an inspiral that can end either with:
- a stellar merger
- the successful ejection of the shared envelope (interrupting the drag), and the formation of a tight period binary
Common envelope evolution, since it’s theorization in the 1970s by Paczynski, Webbink, Taam, and Ostriker, remains one of the biggest open questions in stellar physics that impacts the formation of all compact binaries (cataclysmic variables made of a main sequence plus a white dwarf, binary white dwarfs, gravitational wave mergers, etc.).
Depending on when mass transfer starts, we can have three different categories (defined by Kippenhahn et al. 1967 and Lauterborn 1970):
- case A RLOF: donor is burning hydrogen in its core, thus it occurs in the tightest (smaller separation a) binaries when the stars are still relatively small in radius.
- case B RLOF: the donor has a helium core (possibly inert and sustained by electron degeneracy or burning), typically this is a faster mass, although this may also depend on the metallicity and its impact on where on the HR diagram Helium ignites (see e.g., Klencki et al. 2022). Since stars typically expand after they run out of hydrogen-rich fuel in the core, this mode of mass transfer is the most common overall.
- case C RLOF: for low mass stars, this is typically defined after He ignition, for high mass stars, it is typically defined after core He depletion.
Because of the different L and φ factors for each nuclear burning phase (see lecture on nuclear burning for notation), the timescales for mass transfer and therefore responses of both stars and the orbit to mass transfer can be very different for these three, leading to divergent evolution of the binaries and the outcome: case A typically has a short thermal timescale phase, followed by a much longer nuclear timescale phase. Case B and C tend to be faster (thermal timescale or shorter), though this may also be metallicity dependent .
Because of the evolution of the stellar radius R(t), many stellar binaries will evolve from an initially detached state (i.e., the mass of each star is contained within their respective Roche lobes), to a semi-detached phase (when one star’s material fills and even exceeds its Roche lobe, causing matter to be pulled away by the companion’s gravity), and a fraction even into over-contact (both stars filling their Roche lobes).
During a phase of mass transfer the structure of both stars changes, but also the orbit! The details of this are arguably the biggest uncertainty in binary stellar evolution:
- in most cases this process is fast (\propto thermal timescale), making direct observations rare (with the exception of case A RLOF on nuclear timescales)
- where, how much, and how mass is transferred and lost from the system determines the torques that the orbit feels and thus it’s angular momentum and energy evolution
- a variety of physical processes (from magnetic fields, jets, etc.) can intervene in the mass transfer, and with different degrees of importance on the process depending on the masses and mass-ratios of the system, and the evolutionary stage: a general recipe valid for all binaries is not known and may not even exist because of the wide range of physical regimes in which mass transfer could occur!
To study the evolution of the orbit, it is useful to consider the total orbital angular momentum of the binary:
with Ji = Ii ωi spin angular momentum of the two stars, and Jorb orbital angular momentum. A simple order of magnitude calculation can show that
where a\gg{R1, R2} is the semimajor axis of the orbit, much larger than the radii of the stars, and ωi \leq n is the spin frequency of the rotation of the stars which is typically lower than the orbital frequency n = 2π/P and at best equal when tides can synchronize rotation and revolution (e.g., in the Moon+Earth binary!). Note however that there are known exception to this (see for example Britavski et al. 2024), but they don’t invalidate the ordering of Eq. \ref{eq:J_ordering} because of the squared dependence on the lever arm length of the angular momentum: the spin angular momentum of both stars in a binary is typically small compared to the orbital angular momentum.
For the orbital term, using Kepler’s 3rd law and a1M1 = a2M2 = a M1M2/(M1+M2) we can write:
and thus taking a logarithmic derivative obtain:
which allows us to calculate the change in orbital separation a if we know the mass change rates and the orbital angular momentum losses from the system and eccentricity changes. In other words, we need to know the forces and torques exerted by the gas being transferred/lost onto the binary.
For the eccentricity, a common (but questionable and questioned)
approach is to assume that tides pre-mass transfer will circularize
the binary (e=0). While this seems plausible since pre-mass transfer
the donor star is by definition almost as large as it can be, allowing
for tides to torque it, in reality post-mass transfer eccentric
systems are known (see e.g., Eldridge 2009): it is possible that the
timescale for circularization
The orbital angular momentum losses can occur do to a variety of phenomena:
- magnetic braking: if the stars are magnetic (e.g., because of the presence of a convective envelope), expelled gas can remain magnetically tethered to them, providing a long lever arm to that gas to extract angular momentum
- tidal LS coupling: if the system is small enough that tides matter, they can redistribute angular momentum between the spins and orbit. In extreme mass ratio binaries with q=M2/M1 \ll 1/3 this can result in unstable situations where the orbit doesn’t have enough angular momentum to allow for tidal synchronization, leading to a runaway extraction of angular momentum from the orbit and ultimately a merger (this is called a Darwin instability, after the nephew of the more famous Charles Darwin of the evolutionary theory)
- gravitational wave emission: this term is important for very long lived and compact binaries, and can ultimately lead to mergers of compact stellar remnants (white dwarfs, neutron stars, and black holes)
- mass loss: where and how mass may be lost by the system (e.g., if the accretor star does not accept all the mass) can also eat up some orbital angular momentum, and the rate depends on poorly understdood details.
The last term is usually the dominant one, but a general solution would require to treat all of the effects above.
Let’s now consider some cases where we can solve the equation for the orbital evolution.
Consider for simplicity a circular binary (e=0) where the only mass lost is due to stellar winds (no mass exchange).
The winds have velocity vwind,i \geq vesc,i ∼(2GMi/Ri)1/2 for each star, while the orbital velocity of the binary is set by Kepler’s 3rd law, and each individual star has
where the square root term is vorb (i.e., the velocity around the center of mass of the reduced-mass point-mass), and the prefactors come from momentum balance in the center of mass frame of the system.
Thus, taking the ratio, we see that:
where the last comes from a \gg Ri. This means that to a good approximation one can assume winds from the stars to leave the system instantaneously without exerting any torque on the binary. In this case, the material lost to winds will carry the specific angular momentum (per unit mass) of the wind-losing star moving on its orbit:
where in the last we used Jorb, i Mi = Jorb M1M2/(M1+M2) (angular momentum conservation in the center of mass frame) and dMi<0 in both cases. The Eq. \ref{eq:am_balance} reduces to
which since dMi<0 means that in the approximation of fast stellar wind mass loss (w.r.t. to the orbital velocity) and assuming the wind to take the specific orbital angular momentum of the mass-losing star around its orbit and no accretion, then the binary widens and the relative rate at which is widens is equal to the relative rate at which the binary loses mass! Note that the orbital angular momentum is decreasing, and yet the binary widens. This is sometimes referred to as “Jeans mode”.
Another case that allows for analytic consideration is the case of conservative mass transfer, that is
that is all the mass lost by star 1 is accreted by star 2, no spills from the system occurs and thus no orbital angular momentum can be lost.
In this case Eq. \ref{eq:am_balance} becomes:
which tells us that since dM1<0 when star 1 is the one filling its Roche lobe and donating the mass, the orbit initially shrinks, until the condition M2=M1 is reached because of mass transfer, after which further mass loss results in widening of the orbit.
Effectively, what we have found is that conservation of angular momentum imposes that the orbit initially shrinks and then widens once the mass ratio flips from q=M2/M1<1 to q>1.
In general, we can assume that a fraction βRLOF of the mass lost by the donor star ends up on the accretor star (thus a fraction 1-βRLOF is instead lost from the system), and that the non-accreted material takes away a specific angular momentum per unit mass γRLOF. With this parametrization of the uncertain mass transfer efficiency (represented by the poorly known parameter βROLF) and angular momentum losses (represented by γRLOF) we can re-write Eq. \ref{eq:am_balance} for a circular system (e=0) as:
where in general βRLOF and γRLOF are going to be unknown functions of the masses and separation (or period), but they can be analytically specified for some physically relevant cases (alternative but equivalent parametrizations exist, see for example Sobermann et al. 1997).
These parameter may also depend on the formation of an accretion disk around the initially less massive star (as opposed to direct impact of the stream of matter with the star), and possibly of a circumbinary disk (see for example Lubow & Shu 1975).
The artist impression in the video above shows a conservative (no spill over), dynamically stable mass transfer between a 20Mo and a 15Mo star producing a direct impact of the L1 stream with the accretor. Although this is an artist impression (as you can tell from the fact that the stars are resolved), it illustrates several important modifications that will occur in both stars as they go through mass transfer.
The donor will lose (most of) its hydrogen rich envelope, becoming a low opacity, high temperature, small in radius, helium enriched star. This is something that only very massive stars may do if single (and how massive they need to be is a matter of debate because of wind uncertainties).
This means that post-mass transfer, the donor star is the exposed core of the star, which allows a unique opportunity to directly see (with photons!) the interior of a star.
The mass transferred comes with angular momentum and will easily spin up the accretor star, making it fast rotating up to the point where at the equator centrifugal force and radiation pressure balance the gravitational pull. This is what is invoked to explain Be stars in Be-X-ray binaries for example.
Secondly, the matter from the donor may be chemically enriched by nuclear processes in the donor’s core. For example, if the star transfers matter from the outermost layers of the core, it may be CNO-processed! This can change the chemical composition in the accretor structure, and thus its spectrum.
Finally, for massive stars, there is a rejuvenation effect. Since they burn through the CN-NO cycle, εCNO∝ T16 releases a lot of energy in a small hot-enough volume, leading to a steep temperature gradient and thus convection. As the mass of the accretor increases because of accretion itself, \langle T \rangle \propto M by the virial theorem means that the average (and central) temperature have to increase to sustain the increased mass. This leads to a steepening of the temperature gradient in the core, and drives further convection. This “extra” convection takes Hydrogen of the accretor that would not have burned had the star been single, and mixes it in the burning region itself, elongating the lifetime of the star.
Say we have a binary that is starting to transfer mass through the L1 Lagrangian point: how can we decide the mass transfer rate per unit time? This is a crucial question, since the response of both stars to mass transfer will depend on the $dMi/dt$ of both stars (or in other words on how the mass-change timescale $Mi/\dot{Mi}$ compares to the local timescales of the stellar surfaces), and thus whether it remains dynamically stable or devolves into a common envelope!
The classic approach is to postulate that the mass transfer rate is a function of the amount of overflow in radius. If the donor is star 1:
By dimensional analysis we can infer that the mass loss rate by the donor is related to the gas density and velocity at L1 (the point where it becomes gravitationally unbound) times the cross section of the nozzle of gas across the plane perpendicular to the axis connecting the two stars and going through L1
N.B.: this is reminiscent of the time-dependent mass-conservation term we obtained during the lecture on hydrostatic equilibrium, not by accident!
Each term needs to be estimate based on thermo- and hydro-dynamical considerations on the flow of the gas, but typically vL_{1}≥ vesc,1≅ csound,1 because the gas has to leave the gravitational potential well of star 1 and the last step is a consequence of the Virial theorem, ρL_{1} can be estimated using an EOS and mass continuity from the stellar surface to L1, and to estimate AL_{1} we consider that the Roche potential has a saddle point in L1, therefore we can do a Taylor approximation along the y-direction the plane of interest and get:
where ∂xΦ(L1)=∂yΦ(L1)0, and using the Roche potential plus Kepler’s 3rd law this gives:
with n= 2π/P the orbital frequency. AL_{1}≅ y2 for y determined by how much outside of its Roche lobe the donor star is. This can be related by looking at a point far from L1’s direction where the gravity of the companion does not distort the equipotential surfaces too much, resulting in Φ ∼ GM1/(R1), and thus:
and putting things together:
More detailed calculations use the Bernoulli principle applied to the gas streaming from the outer layers of the donor star to the L1 point, and differ in whether the inner point is in the stellar atmosphere (which is optically thin, likely non-isothermal) or inside the photosphere (optically thick), see for instance Ritter 1988, Kolb & Ritter 1990, Marchant et al. 2021, Cehula & Pejcha 2023, Ivanova et al. 2024.
Once the mass loss rate from the donor is determined, several processes occurring during the travel of the gas in between the stars (does it form a disk or not? does the gas have time to cool and change its entropy?) and when it reaches the accretor (what is the shear and entropy contrast with the accretor’s outer layers? Is the composition the same?) will determine the mass accretion rate of the accretor, and enter in the parameters βRLOF, and γRLOF we introduced above while determining the orbital evolution in the general case.
Note that a lot of the physics at play here may happen while the stars are not in gravothermal equilibrium, something that is neglected in rapid binary population synthesis calculations based on pre-computed single star models (where the single stars were always in gravothermal equilibrium themselves), cf. notes on codes.
- if there is matter flowing through L1, can it be convective at that location (Hint: think of what force drives the instability)?
- Assuming that star 1 in a binary is filling its Roche lobe and transferring mass, find a relation between it’s average density and the orbital parameters P and q=M2/M1. Since these are (in some cases) observationally measurable, this relation gives a way to measure the average density of stars during mass transfer!