6 th lecture of Compact Object and Accretion, Master Programme at Leiden Observatory

Transcription

1 6 th lecture of Compact Object and Accretion, Master Programme at Leiden Observatory Accretion 1st class study material: Chapter 1 & 4, accretion power in astrophysics these slides at

2 Accretion luminosity If all the kinetic energy of a gas in-falling at a rate given up at the surface of the star, the the accretion luminosity can be estimated as is WD NS

3 Accretion luminosity BHs do not have a hard surface and despite accretion efficiency is < 1 ~1, the The efficiency depends on the BH spin, because it determines its size : η = 5.7%, no spin; η = 42%, maximally spinning BH; see previous class η Note: nuclear efficiency H--> He η = 0.7 % Risco = 3 Rs, no spin Risco = 1/2 Rs, max spin isco = innermost stable circular orbit

4 reminder from last class orbit: last stable circular orbit for Schwarzschild BH We find that the last stable circular orbit for a Schwarzschild BH has : 3 So the maximum energy that can be liberated per unit time by a non-rotating BH is No accretion disc exists < 3 Rs On this orbit matter has an energy per unit mass, without rest mass energy of

5 Eddington limit radiation pressure force onto an electron: gravitational force onto electron + proton: Accretion is inhibited if Note: approximate limit that assume spherical symmetry, optically thin or radiative transfer of photons

6 Accretion in binary systems

7 Orbital motion We work under assumption of circular orbit, justified for tight binaries, where tidal forces act rapidly, in less then system age We define the mass ration where M1 is the compact object The binary semi-major axis is related to its period by Kepler 3 rd law note: Rsun ~ km

8 Orbital motion angular frequency vector normal to orbital plane distances of two stars from CM

9 The Roche geometry Consider the motion of a test mass in the combined potential of Assumptions: the two stars The two stars spherically symmetric, and can be regarded as point mass dynamically (mass concentrated for most stars) Circular orbit (tides) Synchronous (tides)

10 The Roche geometry Euler equation Euler equation in co-rotating frame There are inertial forces : Coriolis and centrifugal forces Roche effective potential Note, in hydro equilibrium (v=0), the isobars (curve along which P=const) corresponds to curve of constant ϕr (example the surface of a star)

11 The shape is entirely governed by q (here q=0.25) and the overall scaling by the binary distance a Roche potential Main features: Matter at r >> a, sees the system as a point mass at CM (circular curves) Matter at less then a1/a2 around M2, M1, sees only the potential of a star (circular curves) Figure-of eight area: critical surface that defines the Roche lobes The inner Lagrangian point L1, saddle (unstable point) where mass can flow from one star to the other from accretion power in astrophysics

12 The size of the Roche Lobe Radius of sphere with same volume as the lobe q=1 q=0.5 q=0.1

13 Classification Detached Systems: each star is well inside its Roche lobe. No mass exchange (WD detached systems of exercise 2) or accretion of companion wind (HighMassXRBs =HMXRBs) Semi-detached Systems: The companion of the CO fills its Roche Lobe. Masse can flow from M2 to M1 (CVs, LowMassXRBs, some HMXRBs) Contact binaries: both stars fill their Lobe. Does not happen if one is a CO (can happen before the formation of the CO)

14 Semi-detached systems Mass can flow from the companion to the compact object through the L1 point: Roche Lobe overflow accretion

15 Mass/Radius-Period relation an application to lower main sequence stars If star M2 fills its Roche Lobe, its average mass can be estimated as this is similar to a low mass (~0.1 Msun) star density. => Low mass stars with period ~1 h can fill their lobe For stars on low main sequence R ~M, and one obtains an approximate mass-period relation and a radius-period relation note:

16 Mass flux q=0.5

17 mass flux expanding the potential around L1, along y-axis to overflow its thermal energy greater than centrifugal barrier = 1/2 cs 2 the thickness of the stream is thus And the mass flow with cs ~5 km/s (T~ k), ρ~ 100 g cm -3 P

18 fate of the stream note: v~0 at L1, therefore the stream is captured by the potential of the compact object It makes precessing (because of the presence of M2) elliptical orbits, that intersects and energy is dissipated (not angular momentum) b1 tends towards the minimum energy configuration: circular conservation of angular momentum reads circularization radius:

19 remarks circularization within Roche Lobe for any q can be shown that R circ > size compact object circularization always happens and leads to disc formation. The accretion then occurs because of viscous processes (next week)

20 Orbital evolution mass transfer --> change of q --> change of Roche Lobe companion s Roche Lobe total mass total angular momentum (neglecting spins) mass conservation the companion feeds the CO may be lost with mass leaving the system change in orbital separation change in Roche Lobe s size

21 The Roche Lobe decreases It depends on Let s now assume: Results for smaller mass ratios note x<0 for WD x~1 for MS for any change in companion s size In a semi-detached systems initially, then if q > qcrit, the size of companion becomes larger than its RL if q < qcrit, the size of companion becomes smaller than its RL qcrit is found from setting it is close to 5/6. For x=1 and no J loss qcrit = 4/3

22 consequence on mass transfer the case of high mass ratios, q > q crit R2 > Roche Lobe, mass transfer starts and mass transfer increasing keeps On can show that in either a dynamical or a thermal timescale (depending on whether the envelope is radiative or convective) the companion loses a substantial fraction of its mass It is an unstable mass transfer of short duration that stops when q < qcrit This can be hardly observed, to explain the majority of CV or XRB observed, we need a stable mass transfer, that for q <qcrit

23 consequence on mass transfer the case of lower mass ratios, q < q crit R2 < RL, mass transfer inhibited, can restart if: 1) the companion radius increases its radius for stellar evolution process tnuc = nuclear timescale of that particular evolution phase, e.g. red giant phase is less 10% of main sequence lifetime for Sun also the change in separation etc..happens on the same timescale This is not the most frequent case, because tnuc is generally short

24 consequence on mass transfer the case of lower mass ratios, q < q crit and R2 < RL, mass transfer inhibited, can restart if: 2) The Roche Lobe decreases, because of loss of angular momentum For The mass transfer restart when and stops again: it proceeds in steps, which makes it stable The system has therefore always

25 the case of lower mass ratios, q < q crit and P note: β =0.46, from the mean density in Roche Lobe formula The mass transfer, stellar size, and separation proceeds on the timescale for J loss, what is it?

26 Loss of angular momentum 1. Loss for gravitational wave emission high dependence on orbital period: the loss is ~500 times smaller for a binary with P= 1 h, then for P= 10 h, all other parameters being equal Dominant for short period binaries P P I used previous slide s results

27 Loss of angular momentum 1. Loss for magnetic braking Dominant for long period binaries in The dipolar magnetic field of the companion creates a co-rotating magnetosphere, to which the stellar wind is anchored. The corotation implies larger angular momenta at larger distance, this can be transported away by a comparatively small amount of mass The derivation of magnetic breaking would require a detailed derivation, out of the scope and (timescale) of this course

28 Application to CVs A Cataclysmic variable star is a binary containing an accreting white dwarf Companion (M2): 1) A main sequence star 2) A white dwarf

29 Cataclysmic variable Mass-Period relation for M2 P reminder: P P Orbital evolution for mass transfer 1st remark: Different companion have different coefficients, the sign depends on q differently

30 Cataclysmic variable Mass-Period relation for M2 P P Orbital evolution for mass transfer second remark: (mass transfer occurs) if q = M2/M1< 4/3 (MS) & if q < 2/3 (WD) ===>The companion must be smaller for WD-WD and from slightly larger to smaller for WD-MS. I.e. q<~1 Note: qcrit < 4/3 (MS) & <2/3 (WD)

31 Cataclysmic variable Mass-Period relation for M2 P P Orbital evolution for mass transfer 3rd remark: if and WD ) the binary shrinks in the first case and widen in the second

32 Cataclysmic variable Mass-Period relation for M2 P P Orbital evolution for mass transfer 4th remark: We note again that the evolution is on the angular momentum loss timescale.

33 magnetic braking in CVs This mechanism dominates at periods P >3 h : For MS implies this M > 0.3 M sun For a WD that P would imply an unphysical mass P P =0.33 Msun = Msun For M < 0.3 Msun, the star has no longer a radiative core, but has become fully convective. The standard picture is that bipolar magnetic field cannot longer be sustained and the braking mechanism switches off see e.g.: Zangrilli, Tout, Bianchini, 1997, MNRAS, 289,59

34 Period gap Gravitational wave: the period dependence is strong, it becomes the dominant mechanism for P < 2 h Period gap between 2 h < P < 3h paucity of CVs detected with that period, in an otherwise smooth P distribution

35 Mass flux in CVs for GWs P P The numerical factor varies only by a factor of few with q, thus the mass transfer mainly depends on the PERIOD. Note the Period dependence is very different

36 Understanding the period distribution further maximum period: ~12-13 h. From what said, it must be WD-MS with q<1 for stable mass transfer. P if = 1.4 M sun P ~12 h ~3 h < P <~12 h. Magnetic braking. WD-MS. For <3 h (M2 <0.3 Msun) the star becomes fully convective and the mechanism is no longer on ~2 h < P <~3h period gap ~80 min < P <~ 2h. Gravitational wave regime. For such P, M2 can be either a MS (0.15 < M2 <0.22) or a WD. representative sample of CVs from Ritter & Kolb min > P <~ 2h. For such P, M2 cannot be a MS (P= 1 h = 0.1 Msun) it can only be a WD with increasing mass

37 Detached systems: wind mass transfer (briefly) e.g. the High mass X-Ray binaries Cyg X-1, Cen-X-3, Vela X-1

38 wind from massive stars Typical detached system contain M1 = neutron star or BH and a massive star M2 (Type O/B) M2 has typically a mass loss of ~ Msun yr -1 The wind velocity is supersonic (sound speed ~10-12 km/s) and equal to wind wind also faster than the relative velocity between stars: 2πa/P ~210 km/s (1+q) 1/3 (M1/Msun) 1/3 (P/1day) -1/3 => balistic gas impacting onto M1 with v~vwind

39 wind accretion gas with kinetic energy < the gravitational energy get captured by M1, therefore we can estimate the accretion radius as 1/2 v 2 wind - G M1/Racc =0, wind (vwind ~vesc) wind The accretion cross section is π( ) ) 2 the accretion rate is thus a fraction of the mass loss wind For a HMXB with typical P = 1 day, we get: wind

40 wind accretion continue... the small cross section is compensated by large wind loss, and the results are mass accretion rates comparable or larger than those for Roche Lobe overflow ==> HMXR can be very luminous: Circularization radius: At gas has an angular momentum related to the (so far neglected orbital motion): ( Ω) A Keplerian Circular Orbit with same angular momentum has a radius Rcirc can be very small (see table) it is therefore not clear that a disc forms in every case presence of magnetic field force the gas to accrete at poles (see discussion in Frank, King & Raine and lecture later in course)