Lecture 5: Tides

What are tides

This is the fluids response of a star to the companion’s gravity

The tidally distorted star is elongated along the equatorial plane of the binary. The closer/heavier the companion, the stronger the distorion.

There are two types of tides (approx.), which ones?

1) Equilibrium tides:

• Impose hydrostatic equilibrium to approximate the response of the fluid

• Changes pressure and density cond. inside star to satisfy distorted hydrostatic equilibrium (star is treated as continuously adjusting to equilibrium shape)

• Good first order approx.

2) Dynamical tides:

• Corrections to equilibrium approx.

• Accounts for any acceleration and damping (occiliations, internal grav. waves, dissipation mechanisms,…) of the fluid due to it’s displacement

• Can excite internal waves

Why do tidal effects become important in close binaries?

Because the tidal force scales strongly with separation:

F_tidal ∼ R/a³

What are the two types of rotation in a binary system?

• Orbital rotation (motion of the stars around each other)

J_orb = M₁a²Ω_orb + M₂a²Ω_orb

• Stellar rotation (spin of each star around its own axis)

J_rot = I₁Ω₁+I₂Ω₂

Are stars in binaries born with synchronized spins and orbits?

No. Generally:

Ω_orb ≠ Ω₁≠ Ω₂

Can a real star instantaneously match the tidal equilibrium shape?

No. Stars have internal viscosity and finite response times.

What happens if the tidal bulge is misaligned with the line of centers?

A tidal torque acts on the star and the orbit.

What is tidal torque

The torque exerted on a star by it’s companion due to the misalignment between the tidal bulge and the line connecting the two stars, which transfers angular momentum between the stars rotation and binary orbit

It brings the bulge back in alignment by increasing/decreasing the rotational velocity of the distorted star until it matches orbital rotation

Once misalignment is null, torque disappears

What happens when the rotational velocities of the stars (Ω₁,Ω₂) changes

this has to result in changes in orbital angular momentum and/or separation

What is tidal synchronisation

This is when the stellar and orbital rotational velocities are equal

What is tidal circularisation

This is when the orbit becomes circular.

This is because in an eccentric orbit, tidal torques spin the star up near periastron and slow down near apastron.

Energy is dissipated inside the star during this process, and this reduces the contrast between periastron & apastron separations, thereby removing eccentricity and circulising the orbit

On what does the tidal equilibration timescale in a binary system depend?

The tidal equilibration timescale (time to reach spin–orbit synchronization and/or circularization) depends on:

• Initial separation & mass ratio

• stellar structure of distorted star

  • convective or radiative envelope

When do stars with convective envelope (low mass main-sequence stars, evolved giant stars,..) reach pseudo-synchronisation?

around 10⁻³ ζ_circ

When do we consider a system pseudo-synchronised and when circularised?

We generally consider that systems with P<20d are pseudo-synchronised and those with P<10d have circularised

How can we observe tides

There is

• Ellipsoidal variability

• small or zero eccentricity in small period systems

What are the tide implications on stellar rotation

Tides are partly responsible for the difference in rotation rates between binaries and single stars

Important cause stellar rotation impacts evolution!

How does stellar rotation impact evolution

• Higher spin = higher mass loss

• Rotationally induced mixing changes surface abundance

• Increased lifetime due to mixing

What are the two main formation channels (they thought) for binary stars?

• Dynamical formation (capture)

(relevant in dense stellar clusters)

• Stars formed within some protostellar cloud

In what environments is dynamical capture most relevant?

Dense stellar environments such as globular clusters

Once a binary forms, what fundamental conservation law governs its orbital evolution?

Conservation of total angular momentum

M₁a²Ω_orb + M₂a²Ω_orb + I₁Ω₁+I₂Ω₂ = cst

Why does stellar evolution affect the orbit in a binary system?

Stellar evolution changes the stellar radius, since I=MR², the moment of inertia is changed, so the orbit has to adjust to an expanding star by decreasing the orbital angular momentum

What is the “short-period compact binary problem”?

Observed compact binaries have very short orbital periods that cannot be explained by angular momentum loss mechanisms (tidal interactions or gravitational waves) alone

(The questions is what mechanisms can drastically lower angular momentum on short timescales?)

What two broad mechanisms are proposed to explain strong angular momentum loss in binaries?

• Mass transfer

• Mass loss (winds, common envelope evolution, mass flow through L2 or L3)

How do mergers exist

There are different mechanisms for mergers:

• Gravitational waves

• “Failed” common envelope evolution

• Tidal instabilities

• unstable contact binary

What is the result of a merger

It results in various types of transients (Ia supernovae, luminous red novae,…) or exotic merger products (blue/red/yellow stragglers)

What is the Darwin instability in close binaries?

An instability that occurs when the spin angular momentum change of a star becomes too small compared to the change of orbital angular momentum (dΩ_orb/dt > dΩ₁/dt), so the tidal torques cannot bring the system in synchronous rotation.

What happens when there is Darwin instability

As the tides extract angular momentum from the orbit to spin up the star, the orbital separation shrinks and the orbital frequency increases faster than stellar spin can follow, leading to runaway inspiral and eventual merger (or at least undergo a phase of common envelope evolution)

What is the total angular momentum of a binary system?

J_tot = J_orb + J₁+J₂

What is the orbital angular momentum for a circular binary?

J_orb = μa²Ω_orb

with μ=M₁M₂/(M₁+M₂)

What is the spin angular momentum of a star?

Jₙ = IₙΩₙ

How does the stellar moment of inertia scale?

I∼MR²

What is the orbital angular frequency from Kepler’s third law?

Ω_orb = √G(M₁+M₂)/a³

What is the Darwin stability condition?

It is stable when
dΩ_orb/dt < dΩ₁/dt

or

J_orb>3J₁

or

a>a_darwin = √(3I₁/μ)

What can trigger a Darwin instability

If the primary star grows too much it can trigger Darwin instability (since I₁∼M₁R₁²)

What can prevent Darwin instability

If the star overfills it’s Roche lobe before reaching big sizes the Darwin instability cannot occur

How do we know if a system is sensitive to darwin instability

Evaluate a_darwin = √(3I₁/μ) when the primary is at Roche lobe overflow to know if a system is sensitive to this instability