2. Accretion as a Source of Energy PhD Course, University of Padua - PowerPoint PPT Presentation

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri 2. Accretion as a Source of Energy PhD Course, University of Padua Page 1

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri High Energy and Time Resolution Astrophysics High Energy and Time Resolution Astrophysics deals mostly with the Astrophysics of Neutron Stars (NSs) and Black Holes (BHs), and their environment. High Energy (HE) emission and rapid time variability phenomena are “boosted” when compact objects reside in binary systems and interact with their companion stars because of mass transfer. PhD Course, University of Padua Page 2

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri Preliminaries Motion of a test particle of mass m in the gravitational field of a compact object. Order of magnitude estimate of the energy that can be extracted: ∆ E/mc 2 = GM c /c 2 r c = 0 . 15( M c /M ⊙ )( r c / 10 6 cm) − 1 = ⇒ compactness M c /r c (1) � Ldt , L action of a single particle): Hamilton-Jacobi equation ( S = g ij ∂S ∂S ∂x j + m 2 c 2 = 0 (2) ∂x i In the Schwarzschild metric ( ǫ = E/mc 2 , h = L/m , S = − Et + Lφ + S r ( r ) , θ = π/ 2 ): r 2 ˙ c 2 + V eff = ǫ 2 (3) r = p r � ∂S r � m = 1 mg rr ˙ (4) ∂r 1 + h 2 � � � 1 − r S � r S = 2 GM/c 2 V eff = (5) r 2 c 2 r V eff = h 2 / 2 r 2 c 2 − GM/rc 2 in the Newtonian case (6) PhD Course, University of Padua Page 3

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri √ Maximum energy extraction: η = V eff ( ∞ ) 1 / 2 − V eff ( r ISCO ) 1 / 2 = 1 − 2 2 = 0 . 057 3 √ where r ISCO = 3 r S is the innermost stable circular orbit (ISCO) and h ISCO = 3 r S · c . In the Kerr metric: η = 0 . 42 Accretion luminosity ( ˙ M accretion rate, mass accreted per unit time): Mc 2 ≃ 10 38 η 0 . 1 ˙ L acc = η ˙ M − 8 erg / s (7) If this energy is radiated without being thermalized (without that radiation has reached thermodynamic equilibrium with the accreting matter): T ∼ 10 12 K ∼ 100 MeV GMm p /r c ∼ kT = ⇒ (8) If it is radiated as blackbody radiation (matter optically thick): T ∼ 10 7 K ∼ 1 keV L acc = 4 πr 2 c σT 4 = ⇒ (9) For a steadily radiating source and assuming spherical symmetry , the maximum luminosity above which accretion quenches can be obtained setting the radiative force ( F rad = σ T L/ 4 πr 2 ) equal to the gravi- tational force ( F g = GMm p /r 2 ) acting on a electron-proton pair: L Edd = 1 . 3 × 10 38 M 1 erg / s Eddington luminosity (10) PhD Course, University of Padua Page 4

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri Wind and Roche-lobe fed binary systems WIND ACCRETION • Stellar wind from early type companions: v W ∼ 1000 km s − 1 >> v s • Compact object orbital velocity (Kepler’s law): v c ∼ 200 M 1 / 3 ∗ , 1 (1 + M c /M ∗ ) 1 / 3 P − 1 / 3 km s − 1 day c ) 1 / 2 ≈ v W • Wind velocity relative to the compact object: v rel ∼ ( v 2 W + v 2 • Radius at which the gravitational energy equals a particle kinetic energy (accretion radius): r acc = 2 GM/v 2 rel ≃ 2 GM/v 2 (11) W • Fraction of stellar wind captured by the compact object: f = πr 2 4 πa 2 ≃ G 2 M 2 W a 2 = 2 . 1 × 10 − 3 M 4 / 3 W, 1000 P − 4 / 3 acc c c, 1 v − 4 (12) day v 4 where a = ( G/ 4 π 2 ) 1 / 3 M 1 / 3 P 2 / 3 . c • Luminosity produced by wind accretion (assuming matter has sufficient angular momentum to form a disk): M W = f ˙ ˙ M ∗ = 10 − 8 f − 3 ˙ M ∗ , − 5 M ⊙ / yr (13) L acc,W = 10 38 η 0 . 1 ˙ M W, − 8 erg / s (14) M ∗ < 10 − 5 M ⊙ / yr and/or matter does not form a disk, so that η < 0 . 1 . Therefore, wind Often ˙ PhD Course, University of Padua Page 5

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri accretion is inefficient. Only thanks to the high mass loss rates of early type stars, can sources powered in this way be luminous. • Does accreting matter have enough angular momentum to form a disk? The accretion stream rotates around the compact object with angular velocity ω orb = v c /a = 2 π/P : l W ∼ r acc × ( ω orb r acc ) = 2 πr 2 acc /P (15) The angular momentum is low enough that matter accretes directly on the compact object. However, the donor may be very close to filling its Roche lobe (see below) when the compact object is at periastron, allowing the formation of an accretion stream or even a transient accretion disc (e.g. Negueruela 2010). ROCHE-LOBE OVERFLOW Motion of a test particle in the gravitational potential of two massive bodies (restricted three-body problem) in circular orbit about each other. Euler equation in the reference frame corotating with the binary governed by the Roche potential: Φ R = − GM c | r − r ∗ | − 1 GM ∗ 2( ω × r ) 2 | r − r c | − (16) PhD Course, University of Padua Page 6

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri Figure 1: From Frank, King & Rayne (2002, Accretion Power in Astrophysics; left) and https://en.wikipedia.org/wiki/Roche − lobe/media/File:RochePotential − color.PNG (right). If at a certain stage of its evolution the companion star ( M 2 = M ∗ ) swells up so that its surface reaches contact with its Roche lobe R 2 , any small perturbation will push material over the saddle point L 1 (inner Lagrange point) of Φ R where it is eventually captured by the compact object . � ˙ ( − ˙ ˙ � M ∗ ) 1 R 2 J tot = − 2 (17) M ∗ 5 / 3 − 2 q R 2 J tot where q = M ∗ /M c . If ˙ M tot = 0 and ˙ J tot = 0 the mass transfer is conservative . PhD Course, University of Padua Page 7

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri Mass transfer from the donor takes place if ˙ M ∗ < 0 ( − ˙ M ∗ > 0 ). If q < 5 / 6 , this condition implies one or both the following possibilities: ˙ R 2 > 0 (18) R 2 ˙ J tot < 0 for e . g . gravitational radiation , magneto − hydro wind (19) J tot In the first case the Roche lobe radius R 2 must expand. So, to sustain a stable mass transfer, also the radius of the donor r ∗ must expand. This occurs when the companion is steadily burning fuel in the core (e.g. during Main Sequence or when it is burning He in the core during the giant phase): ˙ R 2 ∼ ˙ r ∗ 1 ≈ (20) R 2 r ∗ t nuc M ∗ − ˙ ∼ 10 − 8 M ∗ , 1 t − 1 M ∗ = nuc, 50 M ⊙ / yr (21) (5 / 3 − 2 q ) t nuc where t nuc is the hydrogen nuclear burning timescale. Faster mass transfer on a thermal timescale can occur during certain evolutionary phases of the donor. If q > 5 / 6 (and ˙ M tot = 0 , ˙ J tot = 0 ), mass transfer takes place if ˙ R 2 /R 2 < 0 . The Roche lobe R 2 shrinks down on the donor. Unless the star contracts rapidly, the overflow proceeds on a dynamical or thermal timescale, depending on whether the star’s envelope is convective or radiative. This may eventually lead to a rapid and often violent evolution (unstable mass transfer) that may end up in a PhD Course, University of Padua Page 8

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri common envelope phase . Specific angular momentum of the accreting matter for Roche lobe overflow: l R ∼ R 1 × ( ω orb R 1 ) = 2 πR 2 1 /P >> l W (22) Radius R 0 where l R is equal to the Keplerian angular momentum l K ( R 0 ) (circularization radius): R 0 ≈ 1 P 2 / 3 day R ⊙ >> r c = ⇒ formation of a disk (23) PhD Course, University of Padua Page 9

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri Figure 2: From Frank, King & Rayne (2002, Accretion Power in Astrophysics) If there is differential rotation, because of thermal or turbulent motion stresses are generated between neighbouring radii. The resulting effect is a net ’viscous’ torque ( µ is the linear mass density, see below): G = νµr 2 ( ∂ Ω /∂r ) (24) PhD Course, University of Padua Page 10

High Energy and Time Resolution Astronomy and Astrophysics: 2. Accretion L. Zampieri Molecular viscosity alone is far too weak, as are other mixing mechanisms such as convection and tidal mixing. It was concluded that turbulent mixing was needed to generate the observed degree of viscosity. However, in the absence of magnetic fields no instabilities could be found that would drive such turbulence. A breakthrough came with the discovery that the driving mechanism is the magnetorotational instability in a weekly magnetized plasma (MRI; Balbus & Hawley 1991). Figure 3: From http://mri.pppl.gov/ and https://ay201b.wordpress.com/2011/04/11/the-magnetorotational-instability/ PhD Course, University of Padua Page 11